3260:). In the latter case (i.e., a collocation method), the nonlinear optimization problem may be literally thousands to tens of thousands of variables and constraints. Given the size of many NLPs arising from a direct method, it may appear somewhat counter-intuitive that solving the nonlinear optimization problem is easier than solving the boundary-value problem. It is, however, the fact that the NLP is easier to solve than the boundary-value problem. The reason for the relative ease of computation, particularly of a direct collocation method, is that the NLP is
20:
4424:
3362:
5314:
3560:
4109:
3116:) are solved for and the resulting solution is readily verified to be an extremal trajectory. The disadvantage of indirect methods is that the boundary-value problem is often extremely difficult to solve (particularly for problems that span large time intervals or problems with interior point constraints). A well-known software program that implements indirect methods is BNDSCO.
4999:
3010:
2581:
2757:
4716:
4419:{\displaystyle {\begin{aligned}H&=pu_{t}-{\frac {u_{t}^{2}}{x_{t}}}-\lambda _{t+1}u_{t}\\{\frac {\partial H}{\partial u_{t}}}&=p-\lambda _{t+1}-2{\frac {u_{t}}{x_{t}}}=0\\\lambda _{t+1}-\lambda _{t}&=-{\frac {\partial H}{\partial x_{t}}}=-\left({\frac {u_{t}}{x_{t}}}\right)^{2}\end{aligned}}}
3546:. Again it is infrequent, especially in continuous-time problems, that one obtains the value of the control or the state explicitly. Usually, the strategy is to solve for thresholds and regions that characterize the optimal control and use a numerical solver to isolate the actual choice values in time.
2897:
2197:
The infinite horizon problem (i.e., LQR) may seem overly restrictive and essentially useless because it assumes that the operator is driving the system to zero-state and hence driving the output of the system to zero. This is indeed correct. However the problem of driving the output to a desired
186:
Another related optimal control problem may be to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another related control problem may be to minimize the total monetary cost of completing the
2869:
Optimal control problems are generally nonlinear and therefore, generally do not have analytic solutions (e.g., like the linear-quadratic optimal control problem). As a result, it is necessary to employ numerical methods to solve optimal control problems. In the early years of optimal control
2417:
1404:
3341:
provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous-time problem. Not all discretization methods have this property, even seemingly obvious ones. For instance, using a variable
3272:
which are very popular these days) is significantly larger than the range of problems that can be solved via indirect methods. In fact, direct methods have become so popular these days that many people have written elaborate software programs that employ these methods. In particular, many such
5588:
3244:
5309:{\displaystyle {\begin{aligned}H&=pu(t)-{\frac {u(t)^{2}}{x(t)}}-\lambda (t)u(t)\\{\frac {\partial H}{\partial u}}&=p-\lambda (t)-2{\frac {u(t)}{x(t)}}=0\\{\dot {\lambda }}(t)&=-{\frac {\partial H}{\partial x}}=-\left({\frac {u(t)}{x(t)}}\right)^{2}\end{aligned}}}
2647:
1011:
constraints and thus may not be active (i.e., equal to zero) at the optimal solution. It is also noted that the optimal control problem as stated above may have multiple solutions (i.e., the solution may not be unique). Thus, it is most often the case that any solution
4540:
3084:
3309:. These software tools have increased significantly the opportunity for people to explore complex optimal control problems both for academic research and industrial problems. Finally, it is noted that general-purpose MATLAB optimization environments such as
3123:. In a direct method, the state or the control, or both, are approximated using an appropriate function approximation (e.g., polynomial approximation or piecewise constant parameterization). Simultaneously, the cost functional is approximated as a
3455:. The costate summarizes in one number the marginal value of expanding or contracting the state variable next turn. The marginal value is not only the gains accruing to it next turn but associated with the duration of the program. It is nice when
1515:
2202:
the zero output one is. In fact, it can be proved that this secondary LQR problem can be solved in a very straightforward manner. It has been shown in classical optimal control theory that the LQ (or LQR) optimal control has the feedback form
1948:
1146:
2377:
6301:
Izzo, Dario. "PyGMO and PyKEP: open source tools for massively parallel optimization in astrodynamics (the case of interplanetary trajectory optimization)." Proceed. Fifth
International Conf. Astrodynam. Tools and Techniques, ICATT.
5435:
5702:
3164:
6291:
Vasile M., Bernelli-Zazzera F., Fornasari N., Masarati P., "Design of
Interplanetary and Lunar Missions Combining Low-Thrust and Gravity Assists", Final Report of the ESA/ESOC Study Contract No. 14126/00/D/CS, September
6281:
Gath, P.F., Well, K.H., "Trajectory
Optimization Using a Combination of Direct Multiple Shooting and Collocation", AIAA 2001–4047, AIAA Guidance, Navigation, and Control Conference, Montréal, Québec, Canada, 6–9 August
3484:
can be solved analytically, but usually, the most one can do is describe it sufficiently well that the intuition can grasp the character of the solution and an equation solver can solve numerically for the values.
3005:{\displaystyle {\begin{aligned}{\dot {\textbf {x}}}&={\frac {\partial H}{\partial {\boldsymbol {\lambda }}}}\\{\dot {\boldsymbol {\lambda }}}&=-{\frac {\partial H}{\partial {\textbf {x}}}}\end{aligned}}}
2576:{\displaystyle {\dot {\mathbf {S} }}(t)=-\mathbf {S} (t)\mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} (t)+\mathbf {S} (t)\mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t)-\mathbf {Q} }
1855:
2878:. In an indirect method, the calculus of variations is employed to obtain the first-order optimality conditions. These conditions result in a two-point (or, in the case of a complex problem, a multi-point)
2266:
3015:
455:
3342:
step-size routine to integrate the problem's dynamic equations may generate a gradient which does not converge to zero (or point in the right direction) as the solution is approached. The direct method
2902:
561:
2636:
2001:
1568:
5440:
5004:
4545:
4114:
3169:
649:
1415:
4014:
2752:{\displaystyle \mathbf {0} =-\mathbf {S} \mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} +\mathbf {S} \mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} -\mathbf {Q} }
2182:
4907:
3114:
1866:
1699:
4102:
4992:
3127:. Then, the coefficients of the function approximations are treated as optimization variables and the problem is "transcribed" to a nonlinear optimization problem of the form:
2302:
4711:{\displaystyle {\begin{aligned}\lambda _{t}&=\lambda _{t+1}+{\frac {\left(p-\lambda _{t+1}\right)^{2}}{4}}\\x_{t+1}&=x_{t}{\frac {2-p+\lambda _{t+1}}{2}}\end{aligned}}}
833:
796:
3285:
programming language, optimal control software in MATLAB has become more common. Examples of academically developed MATLAB software tools implementing direct methods include
3159:
2408:
2297:
763:
4479:
5371:
3813:
152:
We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to
4535:
2847:
2825:
2803:
2781:
2142:
2120:
2094:
2072:
2047:
2025:
1662:
1640:
1618:
1596:
6139:
Oberle, H. J. and Grimm, W., "BNDSCO-A Program for the
Numerical Solution of Optimal Control Problems," Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, 1989
5431:
3544:
3515:
3482:
3453:
5594:
3338:
4771:
4744:
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911:
884:
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3731:
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999:
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951:
931:
857:
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in the infinite-horizon case are enforced to ensure that the cost functional remains positive. Furthermore, in order to ensure that the cost function is
1708:
1399:{\displaystyle J={\tfrac {1}{2}}\mathbf {x} ^{\mathsf {T}}(t_{f})\mathbf {S} _{f}\mathbf {x} (t_{f})+{\tfrac {1}{2}}\int _{t_{0}}^{t_{f}}\,\mathrm {d} t}
3815:(the cost of extraction increasing with the square of the extraction speed and the inverse of the amount of ore left) and sells ore at a constant price
3248:
Depending upon the type of direct method employed, the size of the nonlinear optimization problem can be quite small (e.g., as in a direct shooting or
172:. For example, the amount of available fuel might be limited, the accelerator pedal cannot be pushed through the floor of the car, speed limits, etc.
2206:
193:
6691:
6639:
5583:{\displaystyle {\begin{aligned}{\dot {\lambda }}(t)&=-{\frac {(p-\lambda (t))^{2}}{4}}\\u(t)&=x(t){\frac {p-\lambda (t)}{2}}\end{aligned}}}
175:
A proper cost function will be a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and
6743:
6428:
464:
3239:{\displaystyle {\begin{aligned}\mathbf {g} (\mathbf {z} )&=\mathbf {0} \\\mathbf {h} (\mathbf {z} )&\leq \mathbf {0} \end{aligned}}}
2588:
1953:
1520:
2857:(or positive semi-definite) solution is the one that is used to compute the feedback gain. The LQ (LQR) problem was elegantly solved by
46:
is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a
5963:
Ross, I. M.; Proulx, R. J.; Karpenko, M. (6 May 2020). "An
Optimal Control Theory for the Traveling Salesman Problem and Its Variants".
569:
3617:
Consider the problem of a mine owner who must decide at what rate to extract ore from their mine. They own rights to the ore from date
142:
3914:
4801:
3575:. In particular, the law of evolution mentioned in the example is not mentioned in the article and is probably not the same as
3313:
have made coding complex optimal control problems significantly easier than was previously possible in languages such as C and
6192:
6111:
5939:
5868:
2585:
For the finite horizon LQ problem, the
Riccati equation is integrated backward in time using the terminal boundary condition
3268:) to solve large sparse NLPs. As a result, the range of problems that can be solved via direct methods (particularly direct
6802:
4045:
6222:
User's Guide for DIRCOL (version 2.1): A Direct
Collocation Method for the Numerical Solution of Optimal Control Problems
4941:
6417:
PROPT – MATLAB Optimal
Control Software," 1260 S.E. Bishop Blvd Ste E, Pullman, WA 99163, USA: Tomlab Optimization, Inc.
3855:
cannot be sold and has no value (there is no "scrap value"). The owner chooses the rate of extraction varying with time
3517:, the turn-t optimal value for the control can usually be solved as a differential equation conditional on knowledge of
133:
describing the paths of the control variables that minimize the cost function. The optimal control can be derived using
6845:
6224:, Fachgebiet Simulation und Systemoptimierung (SIM), Technische Universität Darmstadt (2000, Version of November 1999).
3079:{\displaystyle H=F+{\boldsymbol {\lambda }}^{\mathsf {T}}{\textbf {f}}-{\boldsymbol {\mu }}^{\mathsf {T}}{\textbf {h}}}
6808:
5986:"A Nonsmooth Calculus for Solving Some Graph-Theoretic Control Problems**This research was sponsored by the U.S. Navy"
6705:
6681:
6653:
6625:
6597:
6247:
Hargraves, C. R.; Paris, S. W. (1987). "Direct
Trajectory Optimization Using Nonlinear Programming and Collocation".
5912:
3604:
3409:
3133:
657:
134:
4451:
3391:
5341:
3383:
1705:). The LQR problem is stated as follows. Minimize the infinite horizon quadratic continuous-time cost functional
35:
6755:
6442:
On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems
2882:. This boundary-value problem actually has a special structure because it arises from taking the derivative of a
6314:
2151:
6765:
6749:
6399:, described in Rao, A. V., Benson, D. A., Huntington, G. T., Francolin, C., Darby, C. L., and Patterson, M. A.,
6153:
5806:
3387:
3253:
3422:
A common solution strategy in many optimal control problems is to solve for the costate (sometimes called the
3097:
1671:
5725:
3119:
The approach that has risen to prominence in numerical optimal control since the 1980s is that of so-called
2049:
are positive semi-definite and positive definite, respectively. In the infinite-horizon case, however, the
23:
Optimal control problem benchmark (Luus) with an integral objective, inequality, and differential constraint
6392:
2883:
6325:
Theory and
Implementation of Methods based on Runge–Kutta Integration for Solving Optimal Control Problems
6825:
6402:
5750:
6805:– Applications of Optimal Control Theory Using the Pontryagin Maximum Principle with interactive models.
5781:
1510:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),}
1133:
113:
Optimal control deals with the problem of finding a control law for a given system such that a certain
2853:. It is noted that there are in general multiple solutions to the algebraic Riccati equation and the
2190:. Note that the LQ or LQR cost functional can be thought of physically as attempting to minimize the
805:
768:
2382:
2271:
1132:
A special case of the general nonlinear optimal control problem given in the previous section is the
160:
refers specifically to the way in which the driver presses the accelerator and shifts the gears. The
6840:
5904:
5786:
3765:
3582:
3372:
3090:
and in an indirect method, the boundary-value problem is solved (using the appropriate boundary or
180:
169:
6792:
4513:
2830:
2808:
2786:
2764:
2125:
2103:
2077:
2055:
2030:
2008:
1645:
1623:
1601:
1579:
141:
also known as Pontryagin's minimum principle or simply Pontryagin's principle), or by solving the
5832:
5407:
3520:
3491:
3458:
3429:
3376:
1943:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),}
6673:
Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management
5931:
Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management
5735:
3329:
systems and control solutions. In fact, as optimal control solutions are now often implemented
3291:
2879:
1572:
A particular form of the LQ problem that arises in many control system problems is that of the
122:
78:
5755:
2858:
2640:
For the infinite horizon LQR problem, the differential Riccati equation is replaced with the
130:
6431:, 3rd Workshop in Computational Issues in Nonlinear Control, October 8th, 2019, Monterey, CA
5896:
168:
is the minimization of the total traveling time. Control problems usually include ancillary
6530:
6467:
6256:
6041:
5817:
5791:
4749:
4722:
4486:
4019:
3680:
3586:
2372:{\displaystyle \mathbf {K} (t)=\mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t),}
2050:
1668:, the initial time is arbitrarily set to zero, and the terminal time is taken in the limit
889:
862:
146:
114:
5378:
4912:
4781:
3894:
3858:
3736:
3707:
3094:
conditions). The beauty of using an indirect method is that the state and adjoint (i.e.,
8:
6025:
5897:
5745:
3570:
1003:
190:
A more abstract framework goes as follows. Minimize the continuous-time cost functional
138:
71:
50:
with controls corresponding to rocket thrusters, and the objective might be to reach the
6534:
6471:
6260:
6045:
5375:
Using the above equations, it is easy to solve for the differential equations governing
6564:
6499:
5964:
5827:
5321:
4431:
3838:
3818:
3660:
3640:
3620:
3257:
3249:
2891:
984:
964:
936:
916:
842:
652:
43:
6723:
6054:
6029:
2874:
1950s to 1980s) the favored approach for solving optimal control problems was that of
1015:
6701:
6677:
6649:
6621:
6593:
6585:
6556:
6491:
6483:
6328:
6188:
6107:
6007:
5945:
5935:
5908:
5874:
5864:
2096:
are not only positive-semidefinite and positive-definite, respectively, but are also
176:
94:
6169:
6607:
6546:
6538:
6503:
6475:
6380:, Institute of Flight System Dynamics, Technical University of Munich, October 2019
6264:
6165:
6081:
6069:
6049:
5997:
5892:
5720:
5715:
5697:{\displaystyle x(t)={\frac {\left(4-pt+pT\right)^{2}}{\left(4+pT\right)^{2}}}x_{0}}
2887:
2411:
1007:, respectively. Furthermore, it is noted that the path constraints are in general
98:
39:
6568:
6376:, described in Rieck, M., Bittner, M., GrĂĽter, B., Diepolder, J., and Piprek, P.,
5590:
and using the initial and turn-T conditions, the functions can be solved to yield
6796:
6695:
6671:
6643:
6635:
6615:
6396:
6318:
6209:
User's Manual for SNOPT Version 7: Software for Large-Scale Nonlinear Programming
6149:
6099:
6002:
5985:
5811:
5730:
3326:
2186:
118:
90:
67:
6824:
Lecture Recordings and Script by Prof. Moritz Diehl, University of Freiburg on
6667:
5796:
102:
86:
82:
31:
6819:
6814:
2761:
Understanding that the ARE arises from infinite horizon problem, the matrices
6834:
6663:
6560:
6487:
6455:
6401:
User's Manual for GPOPS: A MATLAB Package for Dynamic Optimization Using the
6011:
5949:
5878:
5776:
5770:
3334:
63:
6479:
6332:
6728:
6495:
3884:
to maximize profits over the period of ownership with no time discounting.
3423:
3330:
59:
54:
with minimum fuel expenditure. Or the dynamical system could be a nation's
6760:
6237:, Boeing Information and Support Services, Seattle, Washington, July 1997
6130:. Transactions of the ASME, Journal of Basic Engineering, 82:34–45, 1960
6551:
6519:"Convergence of the Costates Does Not Imply Convergence of the Control"
5740:
47:
19:
6311:
6085:
6780:
CasADi – Free and open source symbolic framework for optimal control
6611:
6518:
5929:
3762:
that the mine owner extracts it. The mine owner extracts ore at cost
3576:
3344:
6752:– Nonlinear Programming, Calculus of Variations and Optimal Control.
6542:
6268:
5984:
Ross, Isaac M.; Karpenko, Mark; Proulx, Ronald J. (1 January 2016).
3361:
3281:, DITAN. and PyGMO/PyKEP. In recent years, due to the advent of the
16:
Mathematical way of attaining a desired output from a dynamic system
6349:
6154:"A Review of Pseudospectral Optimal Control: From Theory to Flight"
5969:
5760:
1850:{\displaystyle J={\tfrac {1}{2}}\int _{0}^{\infty }\,\mathrm {d} t}
6185:
Practical Methods for Optimal Control Using Nonlinear Programming
3314:
55:
5992:. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016.
6389:
5765:
3310:
3282:
2261:{\displaystyle \mathbf {u} (t)=-\mathbf {K} (t)\mathbf {x} (t)}
2005:
In the finite-horizon case the matrices are restricted in that
93:
in the 1950s, after contributions to calculus of variations by
5773:(Modelica-based open source platform for dynamic optimization)
3333:, contemporary control theory is now primarily concerned with
450:{\displaystyle J:=E\,+\int _{t_{0}}^{t_{f}}F\,\,\mathrm {d} t}
6766:
GESOP – Graphical Environment for Simulation and OPtimization
6617:
Applied Optimal Control: Optimization, Estimation and Control
6347:
Enhancements to the DIDO Optimal Control Toolbox, arXiv 2020.
5822:
5801:
3305:
3278:
3265:
6373:
6211:, University of California, San Diego Report, 24 April 2007
5318:
As the mine owner does not value the ore remaining at time
4428:
As the mine owner does not value the ore remaining at time
51:
6779:
6775:
GPOPS-II – General-Purpose MATLAB Optimal Control Software
6128:
A new approach to linear filtering and prediction problems
6456:"A Roadmap for Optimal Control: The Right Way to Commute"
3303:
while an example of an industry developed MATLAB tool is
81:, and is a mathematical optimization method for deriving
6187:(2nd ed.). Philadelphia, Pennsylvania: SIAM Press.
3704:
ore in the ground, and the time-dependent amount of ore
2864:
859:
is the independent variable (generally speaking, time),
556:{\displaystyle {\dot {\textbf {x}}}(t)={\textbf {f}}\,,}
6789:
4483:
Using the above equations, it is easy to solve for the
187:
trip, given assumed monetary prices for time and fuel.
2148:, the additional restriction is imposed that the pair
1719:
1235:
1157:
1139:. The LQ problem is stated as follows. Minimize the
6784:
6774:
6104:
A Primer on Pontryagin's Principle in Optimal Control
5861:
A primer on Pontryagin's principle in optimal control
5597:
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939:
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865:
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808:
771:
660:
572:
467:
196:
70:. A dynamical system may also be introduced to embed
4016:
subject to the law of motion for the state variable
3348:
is based on the Theory of Consistent Approximation.
2631:{\displaystyle \mathbf {S} (t_{f})=\mathbf {S} _{f}}
1996:{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}
1563:{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}
457:
subject to the first-order dynamic constraints (the
156:
the total traveling time? In this example, the term
6620:(Revised ed.). New York: John Wiley and Sons.
6517:Fahroo, Fariba; Ross, I. Michael (September 2008).
3264:and many well-known software programs exist (e.g.,
5983:
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5696:
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2019:
1995:
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757:
644:{\displaystyle {\textbf {h}}\,\leq {\textbf {0}},}
643:
555:
449:
183:are often interchangeable with the cost function.
6721:
6207:Gill, P. E., Murray, W. M., and Saunders, M. A.,
5927:
4719:and using the initial and turn-T conditions, the
2414:. The differential Riccati equation is given as
6832:
6034:Journal of Computational and Applied Mathematics
3320:
74:within the framework of optimal control theory.
6327:(Ph.D.). University of California at Berkeley.
164:consists of both the car and the road, and the
6662:
6246:
6634:
961:respectively. In the calculus of variations,
6645:Deterministic and Stochastic Optimal Control
6406:, University of Florida Report, August 2008.
6148:
5903:. New York: John Wiley & Sons. pp.
3585:. There might be a discussion about this on
5802:PROPT (Optimal Control Software for MATLAB)
4009:{\displaystyle \Pi =\sum _{t=0}^{T-1}\left}
3733:left in the ground declines at the rate of
3390:. Unsourced material may be challenged and
2299:is a properly dimensioned matrix, given as
2177:{\displaystyle (\mathbf {A} ,\mathbf {B} )}
1127:
85:. The method is largely due to the work of
6761:GEKKO - Python package for optimal control
6523:Journal of Guidance, Control, and Dynamics
6516:
6460:Annals of the New York Academy of Sciences
6249:Journal of Guidance, Control, and Dynamics
5891:
1701:(this last assumption is what is known as
1001:are referred to as the Mayer term and the
117:is achieved. A control problem includes a
6584:
6550:
6142:
6053:
6001:
5968:
4902:{\displaystyle \Pi =\int _{0}^{T}\leftdt}
3605:Learn how and when to remove this message
3410:Learn how and when to remove this message
1838:
1387:
1278:
580:
505:
501:
438:
390:
273:
6606:
6322:
6072:(1996). "Optimal Control—1950 to 1985".
5863:. San Francisco: Collegiate Publishers.
4996:Form the Hamiltonian and differentiate:
4746:series can be solved explicitly, giving
4106:Form the Hamiltonian and differentiate:
3256:) or may be quite large (e.g., a direct
3109:{\displaystyle {\boldsymbol {\lambda }}}
1694:{\displaystyle t_{f}\rightarrow \infty }
18:
6785:PROPT – MATLAB Optimal Control Software
6697:Optimal Control Theory: An Introduction
6676:(Second ed.). New York: Elsevier.
6590:Dynamic Programming and Optimal Control
6024:
3102:
3057:
3033:
2955:
2941:
1576:(LQR) where all of the matrices (i.e.,
77:Optimal control is an extension of the
6833:
6820:Pseudospectral optimal control: Part 2
6815:Pseudospectral optimal control: Part 1
6790:OpenOCL – Open Optimal Control Library
6756:DIDO - MATLAB tool for optimal control
6068:
3337:systems and solutions. The Theory of
3063:
3039:
2730:
2683:
2545:
2480:
2346:
1801:
1756:
1341:
1287:
1176:
6362:User's Guide to DIRECT, Version 2.00,
6235:Sparse Optimal Control Software, SOCS
6182:
3835:. Any ore left in the ground at time
3161:subject to the algebraic constraints
2865:Numerical methods for optimal control
62:; the controls in this case could be
6690:
6453:
6098:
5858:
5854:
5852:
5850:
5848:
4097:{\displaystyle x_{t+1}-x_{t}=-u_{t}}
3553:
3388:adding citations to reliable sources
3355:
2410:is the solution of the differential
2100:. These additional restrictions on
6700:. Englewood Cliffs: Prentice-Hall.
4987:{\displaystyle {\dot {x}}(t)=-u(t)}
3071:
3047:
2990:
2909:
1051:
1025:
811:
774:
712:
673:
663:
633:
605:
586:
575:
527:
508:
496:
472:
415:
396:
318:
279:
224:
205:
125:of state and control variables. An
97:. Optimal control can be seen as a
42:over a period of time such that an
13:
6578:
6444:Math. Prog. 62 pp. 385–415 (1993).
5240:
5232:
5116:
5108:
4805:
4785:
4351:
4343:
4216:
4208:
3918:
3898:
2985:
2977:
2937:
2929:
1840:
1740:
1688:
1389:
1120:to the optimal control problem is
440:
14:
6857:
6715:
5845:
3325:The examples thus far have shown
913:is the terminal time. The terms
108:
58:, with the objective to minimize
6350:https://arxiv.org/abs/2004.13112
6233:Betts, J.T. and Huffman, W. P.,
3558:
3360:
3228:
3213:
3205:
3196:
3181:
3173:
3144:
2835:
2813:
2791:
2769:
2745:
2737:
2724:
2709:
2703:
2698:
2690:
2677:
2668:
2663:
2652:
2644:Riccati equation (ARE) given as
2618:
2593:
2569:
2552:
2539:
2524:
2518:
2504:
2487:
2474:
2465:
2451:
2425:
2387:
2353:
2340:
2325:
2307:
2276:
2245:
2231:
2211:
2194:(measured as a quadratic form).
2167:
2159:
2130:
2108:
2082:
2060:
2035:
2013:
1983:
1958:
1924:
1919:
1902:
1897:
1874:
1863:first-order dynamic constraints
1822:
1817:
1795:
1777:
1772:
1750:
1650:
1628:
1606:
1584:
1550:
1525:
1491:
1477:
1460:
1446:
1423:
1412:first-order dynamic constraints
1371:
1357:
1335:
1317:
1303:
1281:
1211:
1200:
1170:
1143:continuous-time cost functional
828:{\displaystyle {\textbf {u}}(t)}
791:{\displaystyle {\textbf {x}}(t)}
143:Hamilton–Jacobi–Bellman equation
6722:Victor M. Becerra, ed. (2008).
6510:
6447:
6434:
6421:
6415:Rutquist, P. and Edvall, M. M,
6409:
6383:
6367:
6354:
6339:
6305:
6295:
6285:
6275:
6240:
6227:
6214:
6201:
6176:
6170:10.1016/j.arcontrol.2012.09.002
6133:
5899:Introduction to Dynamic Systems
3154:{\displaystyle F(\mathbf {z} )}
2403:{\displaystyle \mathbf {S} (t)}
2292:{\displaystyle \mathbf {K} (t)}
758:{\displaystyle {\textbf {e}}=0}
6454:Ross, I M. (1 December 2005).
6120:
6092:
6062:
6018:
5977:
5956:
5921:
5885:
5807:Pseudospectral optimal control
5607:
5601:
5567:
5561:
5546:
5540:
5527:
5521:
5499:
5495:
5489:
5477:
5461:
5455:
5420:
5414:
5391:
5385:
5354:
5348:
5286:
5280:
5272:
5266:
5216:
5210:
5182:
5176:
5168:
5162:
5147:
5141:
5098:
5092:
5086:
5080:
5068:
5062:
5048:
5041:
5029:
5023:
4981:
4975:
4963:
4957:
4925:
4919:
4882:
4876:
4862:
4855:
4843:
4837:
4474:{\displaystyle \lambda _{T}=0}
3871:
3865:
3802:
3796:
3779:
3772:
3749:
3743:
3720:
3714:
3549:
3533:
3527:
3504:
3498:
3471:
3465:
3442:
3436:
3254:pseudospectral optimal control
3217:
3209:
3185:
3177:
3148:
3140:
2610:
2597:
2562:
2556:
2514:
2508:
2497:
2491:
2461:
2455:
2441:
2435:
2397:
2391:
2363:
2357:
2317:
2311:
2286:
2280:
2255:
2249:
2241:
2235:
2221:
2215:
2171:
2155:
1975:
1962:
1934:
1928:
1912:
1906:
1890:
1884:
1835:
1832:
1826:
1813:
1807:
1787:
1781:
1768:
1762:
1745:
1685:
1542:
1529:
1501:
1495:
1487:
1481:
1470:
1464:
1456:
1450:
1439:
1433:
1384:
1381:
1375:
1367:
1361:
1353:
1347:
1327:
1321:
1313:
1307:
1299:
1293:
1275:
1228:
1215:
1195:
1182:
1107:
1068:
1062:
1042:
1036:
1019:
822:
816:
785:
779:
746:
730:
717:
691:
678:
668:
625:
616:
610:
597:
591:
581:
547:
538:
532:
519:
513:
502:
488:
482:
435:
426:
420:
407:
401:
391:
352:
336:
323:
297:
284:
274:
264:
235:
229:
216:
210:
200:
135:Pontryagin's maximum principle
1:
6744:Computational Optimal Control
6429:Computational Optimal Control
6074:IEEE Control Systems Magazine
6055:10.1016/S0377-0427(00)00418-0
5839:
5726:Bellman pseudospectral method
5366:{\displaystyle \lambda (T)=0}
4778:The manager maximizes profit
3891:The manager maximizes profit
3808:{\displaystyle u(t)^{2}/x(t)}
3321:Discrete-time optimal control
2871:
6750:Automatic Control Laboratory
6003:10.1016/j.ifacol.2016.10.208
4530:{\displaystyle \lambda _{t}}
2842:{\displaystyle \mathbf {R} }
2820:{\displaystyle \mathbf {Q} }
2798:{\displaystyle \mathbf {B} }
2776:{\displaystyle \mathbf {A} }
2198:nonzero level can be solved
2137:{\displaystyle \mathbf {R} }
2115:{\displaystyle \mathbf {Q} }
2089:{\displaystyle \mathbf {R} }
2067:{\displaystyle \mathbf {Q} }
2042:{\displaystyle \mathbf {R} }
2020:{\displaystyle \mathbf {Q} }
1657:{\displaystyle \mathbf {R} }
1635:{\displaystyle \mathbf {Q} }
1613:{\displaystyle \mathbf {B} }
1591:{\displaystyle \mathbf {A} }
1137:(LQ) optimal control problem
72:operations research problems
7:
6403:Gauss Pseudospectral Method
5895:(1979). "Optimal Control".
5751:Gauss pseudospectral method
5708:
5426:{\displaystyle \lambda (t)}
3539:{\displaystyle \lambda (t)}
3510:{\displaystyle \lambda (t)}
3477:{\displaystyle \lambda (t)}
3448:{\displaystyle \lambda (t)}
3351:
10:
6862:
6364:Melbourne, Australia, 2008
5928:Kamien, Morton I. (2013).
5782:Linear-quadratic regulator
1950:and the initial condition
1574:linear quadratic regulator
1517:and the initial condition
34:that deals with finding a
6846:Mathematical optimization
6826:Numerical Optimal Control
6158:Annual Reviews in Control
6106:. Collegiate Publishers.
4909:where the state variable
3339:Consistent Approximations
886:is the initial time, and
5787:Model Predictive Control
4776:Continuous-time version
3252:method), moderate (e.g.
1128:Linear quadratic control
6480:10.1196/annals.1370.015
6323:Schwartz, Adam (1996).
6152:; Karpenko, M. (2012).
5833:Trajectory optimization
2886:. Thus, the resulting
6648:. New York: Springer.
5934:. Dover Publications.
5698:
5584:
5427:
5398:
5367:
5332:
5310:
4988:
4932:
4903:
4792:
4767:
4740:
4712:
4531:
4504:
4475:
4442:
4420:
4098:
4037:
4010:
3950:
3905:
3889:Discrete-time version
3878:
3849:
3829:
3809:
3756:
3727:
3698:
3671:
3651:
3631:
3540:
3511:
3478:
3449:
3240:
3155:
3110:
3080:
3006:
2880:boundary-value problem
2843:
2821:
2799:
2777:
2753:
2632:
2577:
2404:
2373:
2293:
2262:
2178:
2138:
2116:
2090:
2068:
2043:
2021:
1997:
1944:
1851:
1695:
1658:
1636:
1614:
1592:
1564:
1511:
1400:
1114:
995:
975:
947:
927:
907:
880:
853:
829:
792:
759:
645:
557:
451:
131:differential equations
79:calculus of variations
28:Optimal control theory
24:
6795:20 April 2019 at the
6748:Dr. Benoît CHACHUAT:
6378:FALCON.m - User Guide
6183:Betts, J. T. (2010).
5756:Generalized filtering
5699:
5585:
5428:
5399:
5368:
5333:
5311:
4989:
4933:
4904:
4793:
4768:
4766:{\displaystyle u_{t}}
4741:
4739:{\displaystyle x_{t}}
4713:
4532:
4505:
4503:{\displaystyle x_{t}}
4476:
4443:
4421:
4099:
4038:
4036:{\displaystyle x_{t}}
4011:
3924:
3906:
3879:
3850:
3830:
3810:
3757:
3728:
3699:
3697:{\displaystyle x_{0}}
3672:
3652:
3632:
3541:
3512:
3479:
3450:
3241:
3156:
3111:
3088:augmented Hamiltonian
3081:
3007:
2844:
2822:
2800:
2778:
2754:
2633:
2578:
2405:
2374:
2294:
2263:
2179:
2139:
2117:
2091:
2069:
2044:
2022:
1998:
1945:
1861:linear time-invariant
1852:
1696:
1659:
1637:
1615:
1593:
1565:
1512:
1401:
1115:
996:
976:
948:
928:
908:
906:{\displaystyle t_{f}}
881:
879:{\displaystyle t_{0}}
854:
830:
793:
760:
646:
558:
452:
22:
6395:24 July 2011 at the
6317:16 July 2011 at the
5893:Luenberger, David G.
5859:Ross, Isaac (2015).
5818:Sliding mode control
5792:Overtaking criterion
5595:
5436:
5408:
5397:{\displaystyle u(t)}
5379:
5342:
5322:
5000:
4942:
4938:evolves as follows:
4931:{\displaystyle x(t)}
4913:
4802:
4791:{\displaystyle \Pi }
4782:
4750:
4723:
4541:
4514:
4487:
4452:
4432:
4110:
4046:
4020:
3915:
3904:{\displaystyle \Pi }
3895:
3877:{\displaystyle u(t)}
3859:
3839:
3819:
3766:
3755:{\displaystyle u(t)}
3737:
3726:{\displaystyle x(t)}
3708:
3681:
3661:
3641:
3621:
3571:confusing or unclear
3521:
3492:
3459:
3430:
3384:improve this section
3277:, SOCS, OTIS, GESOP/
3165:
3134:
3098:
3016:
2898:
2831:
2809:
2787:
2765:
2648:
2589:
2418:
2383:
2303:
2272:
2207:
2152:
2126:
2104:
2078:
2056:
2031:
2009:
1954:
1867:
1709:
1672:
1646:
1624:
1602:
1580:
1521:
1416:
1147:
1016:
985:
965:
937:
917:
890:
863:
843:
806:
769:
658:
570:
465:
194:
166:optimality criterion
147:sufficient condition
115:optimality criterion
6592:. Belmont: Athena.
6535:2008JGCD...31.1492F
6472:2005NYASA1065..210R
6261:1987JGCD...10..338H
6046:2000JCoAM.124..361S
5746:Dynamic programming
4825:
4159:
3988:
3583:clarify the section
3270:collocation methods
1744:
1274:
1106:
1088:
653:endpoint conditions
386:
139:necessary condition
6809:On Optimal Control
5828:Stochastic control
5694:
5580:
5578:
5423:
5394:
5363:
5328:
5306:
5304:
4984:
4928:
4899:
4811:
4788:
4763:
4736:
4708:
4706:
4527:
4500:
4471:
4438:
4416:
4414:
4145:
4094:
4033:
4006:
3974:
3901:
3874:
3845:
3825:
3805:
3752:
3723:
3694:
3667:
3647:
3627:
3536:
3507:
3474:
3445:
3258:collocation method
3250:quasilinearization
3236:
3234:
3151:
3106:
3076:
3002:
3000:
2892:Hamiltonian system
2839:
2817:
2795:
2773:
2749:
2628:
2573:
2400:
2369:
2289:
2258:
2174:
2134:
2112:
2086:
2064:
2039:
2017:
1993:
1940:
1847:
1730:
1728:
1691:
1654:
1632:
1610:
1588:
1560:
1507:
1396:
1246:
1244:
1166:
1122:locally minimizing
1110:
1092:
1074:
991:
971:
943:
923:
903:
876:
849:
825:
788:
755:
641:
553:
447:
358:
177:initial conditions
44:objective function
25:
6724:"Optimal control"
6194:978-0-89871-688-7
6113:978-0-9843571-0-9
6086:10.1109/37.506395
6030:"Optimal Control"
6026:Sargent, R. W. H.
5990:IFAC-PapersOnLine
5941:978-1-306-39299-0
5870:978-0-9843571-0-9
5682:
5574:
5512:
5452:
5331:{\displaystyle T}
5290:
5247:
5207:
5186:
5123:
5072:
4954:
4886:
4702:
4628:
4441:{\displaystyle T}
4400:
4365:
4289:
4230:
4170:
3999:
3848:{\displaystyle T}
3828:{\displaystyle p}
3670:{\displaystyle 0}
3650:{\displaystyle T}
3630:{\displaystyle 0}
3615:
3614:
3607:
3420:
3419:
3412:
3273:programs include
3073:
3049:
2996:
2992:
2961:
2946:
2916:
2911:
2855:positive definite
2432:
1881:
1727:
1430:
1243:
1165:
1053:
1027:
994:{\displaystyle F}
974:{\displaystyle E}
946:{\displaystyle F}
926:{\displaystyle E}
852:{\displaystyle t}
813:
776:
714:
675:
665:
635:
607:
588:
577:
529:
510:
498:
479:
474:
417:
398:
320:
281:
226:
207:
95:Edward J. McShane
6853:
6801:Elmer G. Wiens:
6740:
6738:
6736:
6711:
6687:
6659:
6631:
6603:
6586:Bertsekas, D. P.
6573:
6572:
6554:
6529:(5): 1492–1497.
6514:
6508:
6507:
6451:
6445:
6438:
6432:
6425:
6419:
6413:
6407:
6387:
6381:
6371:
6365:
6358:
6352:
6343:
6337:
6336:
6309:
6303:
6299:
6293:
6289:
6283:
6279:
6273:
6272:
6244:
6238:
6231:
6225:
6218:
6212:
6205:
6199:
6198:
6180:
6174:
6173:
6146:
6140:
6137:
6131:
6126:Kalman, Rudolf.
6124:
6118:
6117:
6096:
6090:
6089:
6066:
6060:
6059:
6057:
6040:(1–2): 361–371.
6022:
6016:
6015:
6005:
5981:
5975:
5974:
5972:
5960:
5954:
5953:
5925:
5919:
5918:
5902:
5889:
5883:
5882:
5856:
5721:Bellman equation
5716:Active inference
5703:
5701:
5700:
5695:
5693:
5692:
5683:
5681:
5680:
5675:
5671:
5652:
5651:
5646:
5642:
5614:
5589:
5587:
5586:
5581:
5579:
5575:
5570:
5550:
5513:
5508:
5507:
5506:
5475:
5454:
5453:
5445:
5432:
5430:
5429:
5424:
5403:
5401:
5400:
5395:
5372:
5370:
5369:
5364:
5337:
5335:
5334:
5329:
5315:
5313:
5312:
5307:
5305:
5301:
5300:
5295:
5291:
5289:
5275:
5261:
5248:
5246:
5238:
5230:
5209:
5208:
5200:
5187:
5185:
5171:
5157:
5124:
5122:
5114:
5106:
5073:
5071:
5057:
5056:
5055:
5036:
4993:
4991:
4990:
4985:
4956:
4955:
4947:
4937:
4935:
4934:
4929:
4908:
4906:
4905:
4900:
4892:
4888:
4887:
4885:
4871:
4870:
4869:
4850:
4824:
4819:
4797:
4795:
4794:
4789:
4772:
4770:
4769:
4764:
4762:
4761:
4745:
4743:
4742:
4737:
4735:
4734:
4717:
4715:
4714:
4709:
4707:
4703:
4698:
4697:
4696:
4668:
4666:
4665:
4649:
4648:
4629:
4624:
4623:
4618:
4614:
4613:
4612:
4585:
4580:
4579:
4557:
4556:
4536:
4534:
4533:
4528:
4526:
4525:
4509:
4507:
4506:
4501:
4499:
4498:
4480:
4478:
4477:
4472:
4464:
4463:
4447:
4445:
4444:
4439:
4425:
4423:
4422:
4417:
4415:
4411:
4410:
4405:
4401:
4399:
4398:
4389:
4388:
4379:
4366:
4364:
4363:
4362:
4349:
4341:
4329:
4328:
4316:
4315:
4290:
4288:
4287:
4278:
4277:
4268:
4260:
4259:
4231:
4229:
4228:
4227:
4214:
4206:
4200:
4199:
4190:
4189:
4171:
4169:
4168:
4158:
4153:
4144:
4139:
4138:
4103:
4101:
4100:
4095:
4093:
4092:
4077:
4076:
4064:
4063:
4042:
4040:
4039:
4034:
4032:
4031:
4015:
4013:
4012:
4007:
4005:
4001:
4000:
3998:
3997:
3987:
3982:
3973:
3968:
3967:
3949:
3938:
3910:
3908:
3907:
3902:
3883:
3881:
3880:
3875:
3854:
3852:
3851:
3846:
3834:
3832:
3831:
3826:
3814:
3812:
3811:
3806:
3792:
3787:
3786:
3761:
3759:
3758:
3753:
3732:
3730:
3729:
3724:
3703:
3701:
3700:
3695:
3693:
3692:
3676:
3674:
3673:
3668:
3656:
3654:
3653:
3648:
3636:
3634:
3633:
3628:
3610:
3603:
3599:
3596:
3590:
3562:
3561:
3554:
3545:
3543:
3542:
3537:
3516:
3514:
3513:
3508:
3488:Having obtained
3483:
3481:
3480:
3475:
3454:
3452:
3451:
3446:
3415:
3408:
3404:
3401:
3395:
3364:
3356:
3299:, FALCON.m, and
3245:
3243:
3242:
3237:
3235:
3231:
3216:
3208:
3199:
3184:
3176:
3160:
3158:
3157:
3152:
3147:
3115:
3113:
3112:
3107:
3105:
3085:
3083:
3082:
3077:
3075:
3074:
3068:
3067:
3066:
3060:
3051:
3050:
3044:
3043:
3042:
3036:
3011:
3009:
3008:
3003:
3001:
2997:
2995:
2994:
2993:
2983:
2975:
2963:
2962:
2954:
2947:
2945:
2944:
2935:
2927:
2918:
2917:
2912:
2907:
2888:dynamical system
2876:indirect methods
2873:
2859:Rudolf E. Kálmán
2848:
2846:
2845:
2840:
2838:
2826:
2824:
2823:
2818:
2816:
2804:
2802:
2801:
2796:
2794:
2782:
2780:
2779:
2774:
2772:
2758:
2756:
2755:
2750:
2748:
2740:
2735:
2734:
2733:
2727:
2721:
2720:
2712:
2706:
2701:
2693:
2688:
2687:
2686:
2680:
2671:
2666:
2655:
2637:
2635:
2634:
2629:
2627:
2626:
2621:
2609:
2608:
2596:
2582:
2580:
2579:
2574:
2572:
2555:
2550:
2549:
2548:
2542:
2536:
2535:
2527:
2521:
2507:
2490:
2485:
2484:
2483:
2477:
2468:
2454:
2434:
2433:
2428:
2423:
2412:Riccati equation
2409:
2407:
2406:
2401:
2390:
2378:
2376:
2375:
2370:
2356:
2351:
2350:
2349:
2343:
2337:
2336:
2328:
2310:
2298:
2296:
2295:
2290:
2279:
2267:
2265:
2264:
2259:
2248:
2234:
2214:
2183:
2181:
2180:
2175:
2170:
2162:
2143:
2141:
2140:
2135:
2133:
2121:
2119:
2118:
2113:
2111:
2095:
2093:
2092:
2087:
2085:
2073:
2071:
2070:
2065:
2063:
2048:
2046:
2045:
2040:
2038:
2026:
2024:
2023:
2018:
2016:
2002:
2000:
1999:
1994:
1992:
1991:
1986:
1974:
1973:
1961:
1949:
1947:
1946:
1941:
1927:
1922:
1905:
1900:
1883:
1882:
1877:
1872:
1856:
1854:
1853:
1848:
1843:
1825:
1820:
1806:
1805:
1804:
1798:
1780:
1775:
1761:
1760:
1759:
1753:
1743:
1738:
1729:
1720:
1703:infinite horizon
1700:
1698:
1697:
1692:
1684:
1683:
1663:
1661:
1660:
1655:
1653:
1641:
1639:
1638:
1633:
1631:
1619:
1617:
1616:
1611:
1609:
1597:
1595:
1594:
1589:
1587:
1569:
1567:
1566:
1561:
1559:
1558:
1553:
1541:
1540:
1528:
1516:
1514:
1513:
1508:
1494:
1480:
1463:
1449:
1432:
1431:
1426:
1421:
1405:
1403:
1402:
1397:
1392:
1374:
1360:
1346:
1345:
1344:
1338:
1320:
1306:
1292:
1291:
1290:
1284:
1273:
1272:
1271:
1261:
1260:
1259:
1245:
1236:
1227:
1226:
1214:
1209:
1208:
1203:
1194:
1193:
1181:
1180:
1179:
1173:
1167:
1158:
1135:linear quadratic
1119:
1117:
1116:
1113:{\displaystyle }
1111:
1105:
1100:
1087:
1082:
1061:
1060:
1055:
1054:
1035:
1034:
1029:
1028:
1000:
998:
997:
992:
980:
978:
977:
972:
952:
950:
949:
944:
932:
930:
929:
924:
912:
910:
909:
904:
902:
901:
885:
883:
882:
877:
875:
874:
858:
856:
855:
850:
834:
832:
831:
826:
815:
814:
797:
795:
794:
789:
778:
777:
764:
762:
761:
756:
745:
744:
729:
728:
716:
715:
706:
705:
690:
689:
677:
676:
667:
666:
650:
648:
647:
642:
637:
636:
609:
608:
590:
589:
579:
578:
565:path constraints
562:
560:
559:
554:
531:
530:
512:
511:
500:
499:
481:
480:
475:
470:
456:
454:
453:
448:
443:
419:
418:
400:
399:
385:
384:
383:
373:
372:
371:
351:
350:
335:
334:
322:
321:
312:
311:
296:
295:
283:
282:
263:
262:
250:
249:
228:
227:
209:
208:
99:control strategy
83:control policies
40:dynamical system
6861:
6860:
6856:
6855:
6854:
6852:
6851:
6850:
6841:Optimal control
6831:
6830:
6803:Optimal Control
6797:Wayback Machine
6734:
6732:
6718:
6708:
6684:
6668:Schwartz, N. L.
6656:
6628:
6600:
6581:
6579:Further reading
6576:
6543:10.2514/1.37331
6515:
6511:
6452:
6448:
6439:
6435:
6426:
6422:
6414:
6410:
6397:Wayback Machine
6388:
6384:
6372:
6368:
6359:
6355:
6344:
6340:
6319:Wayback Machine
6310:
6306:
6300:
6296:
6290:
6286:
6280:
6276:
6269:10.2514/3.20223
6245:
6241:
6232:
6228:
6220:von Stryk, O.,
6219:
6215:
6206:
6202:
6195:
6181:
6177:
6147:
6143:
6138:
6134:
6125:
6121:
6114:
6097:
6093:
6067:
6063:
6023:
6019:
5996:(18): 462–467.
5982:
5978:
5961:
5957:
5942:
5926:
5922:
5915:
5890:
5886:
5871:
5857:
5846:
5842:
5837:
5812:Pursuit-evasion
5731:Brachistochrone
5711:
5706:
5688:
5684:
5676:
5658:
5654:
5653:
5647:
5620:
5616:
5615:
5613:
5596:
5593:
5592:
5577:
5576:
5551:
5549:
5530:
5515:
5514:
5502:
5498:
5476:
5474:
5464:
5444:
5443:
5439:
5437:
5434:
5433:
5409:
5406:
5405:
5380:
5377:
5376:
5343:
5340:
5339:
5323:
5320:
5319:
5303:
5302:
5296:
5276:
5262:
5260:
5256:
5255:
5239:
5231:
5229:
5219:
5199:
5198:
5195:
5194:
5172:
5158:
5156:
5125:
5115:
5107:
5105:
5102:
5101:
5058:
5051:
5047:
5037:
5035:
5010:
5003:
5001:
4998:
4997:
4946:
4945:
4943:
4940:
4939:
4914:
4911:
4910:
4872:
4865:
4861:
4851:
4849:
4830:
4826:
4820:
4815:
4803:
4800:
4799:
4783:
4780:
4779:
4757:
4753:
4751:
4748:
4747:
4730:
4726:
4724:
4721:
4720:
4705:
4704:
4686:
4682:
4669:
4667:
4661:
4657:
4650:
4638:
4634:
4631:
4630:
4619:
4602:
4598:
4591:
4587:
4586:
4584:
4569:
4565:
4558:
4552:
4548:
4544:
4542:
4539:
4538:
4521:
4517:
4515:
4512:
4511:
4494:
4490:
4488:
4485:
4484:
4459:
4455:
4453:
4450:
4449:
4433:
4430:
4429:
4413:
4412:
4406:
4394:
4390:
4384:
4380:
4378:
4374:
4373:
4358:
4354:
4350:
4342:
4340:
4330:
4324:
4320:
4305:
4301:
4298:
4297:
4283:
4279:
4273:
4269:
4267:
4249:
4245:
4232:
4223:
4219:
4215:
4207:
4205:
4202:
4201:
4195:
4191:
4179:
4175:
4164:
4160:
4154:
4149:
4143:
4134:
4130:
4120:
4113:
4111:
4108:
4107:
4088:
4084:
4072:
4068:
4053:
4049:
4047:
4044:
4043:
4027:
4023:
4021:
4018:
4017:
3993:
3989:
3983:
3978:
3972:
3963:
3959:
3955:
3951:
3939:
3928:
3916:
3913:
3912:
3896:
3893:
3892:
3860:
3857:
3856:
3840:
3837:
3836:
3820:
3817:
3816:
3788:
3782:
3778:
3767:
3764:
3763:
3738:
3735:
3734:
3709:
3706:
3705:
3688:
3684:
3682:
3679:
3678:
3662:
3659:
3658:
3642:
3639:
3638:
3622:
3619:
3618:
3611:
3600:
3594:
3591:
3580:
3563:
3559:
3552:
3522:
3519:
3518:
3493:
3490:
3489:
3460:
3457:
3456:
3431:
3428:
3427:
3416:
3405:
3399:
3396:
3381:
3365:
3354:
3327:continuous time
3323:
3233:
3232:
3227:
3220:
3212:
3204:
3201:
3200:
3195:
3188:
3180:
3172:
3168:
3166:
3163:
3162:
3143:
3135:
3132:
3131:
3101:
3099:
3096:
3095:
3070:
3069:
3062:
3061:
3056:
3055:
3046:
3045:
3038:
3037:
3032:
3031:
3017:
3014:
3013:
2999:
2998:
2989:
2988:
2984:
2976:
2974:
2964:
2953:
2952:
2949:
2948:
2940:
2936:
2928:
2926:
2919:
2908:
2906:
2905:
2901:
2899:
2896:
2895:
2867:
2834:
2832:
2829:
2828:
2812:
2810:
2807:
2806:
2790:
2788:
2785:
2784:
2768:
2766:
2763:
2762:
2744:
2736:
2729:
2728:
2723:
2722:
2713:
2708:
2707:
2702:
2697:
2689:
2682:
2681:
2676:
2675:
2667:
2662:
2651:
2649:
2646:
2645:
2622:
2617:
2616:
2604:
2600:
2592:
2590:
2587:
2586:
2568:
2551:
2544:
2543:
2538:
2537:
2528:
2523:
2522:
2517:
2503:
2486:
2479:
2478:
2473:
2472:
2464:
2450:
2424:
2422:
2421:
2419:
2416:
2415:
2386:
2384:
2381:
2380:
2352:
2345:
2344:
2339:
2338:
2329:
2324:
2323:
2306:
2304:
2301:
2300:
2275:
2273:
2270:
2269:
2244:
2230:
2210:
2208:
2205:
2204:
2166:
2158:
2153:
2150:
2149:
2129:
2127:
2124:
2123:
2107:
2105:
2102:
2101:
2081:
2079:
2076:
2075:
2059:
2057:
2054:
2053:
2034:
2032:
2029:
2028:
2012:
2010:
2007:
2006:
1987:
1982:
1981:
1969:
1965:
1957:
1955:
1952:
1951:
1923:
1918:
1901:
1896:
1873:
1871:
1870:
1868:
1865:
1864:
1859:Subject to the
1839:
1821:
1816:
1800:
1799:
1794:
1793:
1776:
1771:
1755:
1754:
1749:
1748:
1739:
1734:
1718:
1710:
1707:
1706:
1679:
1675:
1673:
1670:
1669:
1649:
1647:
1644:
1643:
1627:
1625:
1622:
1621:
1605:
1603:
1600:
1599:
1583:
1581:
1578:
1577:
1554:
1549:
1548:
1536:
1532:
1524:
1522:
1519:
1518:
1490:
1476:
1459:
1445:
1422:
1420:
1419:
1417:
1414:
1413:
1408:Subject to the
1388:
1370:
1356:
1340:
1339:
1334:
1333:
1316:
1302:
1286:
1285:
1280:
1279:
1267:
1263:
1262:
1255:
1251:
1250:
1234:
1222:
1218:
1210:
1204:
1199:
1198:
1189:
1185:
1175:
1174:
1169:
1168:
1156:
1148:
1145:
1144:
1130:
1101:
1096:
1083:
1078:
1056:
1050:
1049:
1048:
1030:
1024:
1023:
1022:
1017:
1014:
1013:
986:
983:
982:
966:
963:
962:
953:are called the
938:
935:
934:
918:
915:
914:
897:
893:
891:
888:
887:
870:
866:
864:
861:
860:
844:
841:
840:
810:
809:
807:
804:
803:
773:
772:
770:
767:
766:
740:
736:
724:
720:
711:
710:
701:
697:
685:
681:
672:
671:
662:
661:
659:
656:
655:
632:
631:
604:
603:
585:
584:
574:
573:
571:
568:
567:
526:
525:
507:
506:
495:
494:
471:
469:
468:
466:
463:
462:
439:
414:
413:
395:
394:
379:
375:
374:
367:
363:
362:
346:
342:
330:
326:
317:
316:
307:
303:
291:
287:
278:
277:
258:
254:
245:
241:
223:
222:
204:
203:
195:
192:
191:
179:of the system.
127:optimal control
119:cost functional
111:
91:Richard Bellman
68:monetary policy
30:is a branch of
17:
12:
11:
5:
6859:
6849:
6848:
6843:
6829:
6828:
6822:
6817:
6812:
6806:
6799:
6787:
6782:
6777:
6769:
6768:
6763:
6758:
6753:
6746:
6741:
6717:
6716:External links
6714:
6713:
6712:
6706:
6688:
6682:
6660:
6654:
6636:Fleming, W. H.
6632:
6626:
6604:
6598:
6580:
6577:
6575:
6574:
6509:
6466:(1): 210–231.
6446:
6433:
6420:
6408:
6382:
6366:
6360:Williams, P.,
6353:
6338:
6304:
6294:
6284:
6274:
6255:(4): 338–342.
6239:
6226:
6213:
6200:
6193:
6175:
6164:(2): 182–197.
6141:
6132:
6119:
6112:
6091:
6061:
6017:
5976:
5955:
5940:
5920:
5913:
5884:
5869:
5843:
5841:
5838:
5836:
5835:
5830:
5825:
5820:
5815:
5809:
5804:
5799:
5797:PID controller
5794:
5789:
5784:
5779:
5774:
5768:
5763:
5758:
5753:
5748:
5743:
5738:
5733:
5728:
5723:
5718:
5712:
5710:
5707:
5705:
5704:
5691:
5687:
5679:
5674:
5670:
5667:
5664:
5661:
5657:
5650:
5645:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5619:
5612:
5609:
5606:
5603:
5600:
5573:
5569:
5566:
5563:
5560:
5557:
5554:
5548:
5545:
5542:
5539:
5536:
5533:
5531:
5529:
5526:
5523:
5520:
5517:
5516:
5511:
5505:
5501:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5473:
5470:
5467:
5465:
5463:
5460:
5457:
5451:
5448:
5442:
5441:
5422:
5419:
5416:
5413:
5393:
5390:
5387:
5384:
5362:
5359:
5356:
5353:
5350:
5347:
5327:
5299:
5294:
5288:
5285:
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3503:
3500:
3497:
3473:
3470:
3467:
3464:
3444:
3441:
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3418:
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3368:
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3350:
3322:
3319:
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3219:
3215:
3211:
3207:
3203:
3202:
3198:
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3191:
3189:
3187:
3183:
3179:
3175:
3171:
3170:
3150:
3146:
3142:
3139:
3121:direct methods
3104:
3092:transversality
3065:
3059:
3054:
3041:
3035:
3030:
3027:
3024:
3021:
2987:
2982:
2979:
2973:
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2960:
2957:
2951:
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2939:
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2915:
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2866:
2863:
2837:
2815:
2793:
2771:
2747:
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2705:
2700:
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2679:
2674:
2670:
2665:
2661:
2658:
2654:
2625:
2620:
2615:
2612:
2607:
2603:
2599:
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2571:
2567:
2564:
2561:
2558:
2554:
2547:
2541:
2534:
2531:
2526:
2520:
2516:
2513:
2510:
2506:
2502:
2499:
2496:
2493:
2489:
2482:
2476:
2471:
2467:
2463:
2460:
2457:
2453:
2449:
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2443:
2440:
2437:
2431:
2427:
2399:
2396:
2393:
2389:
2368:
2365:
2362:
2359:
2355:
2348:
2342:
2335:
2332:
2327:
2322:
2319:
2316:
2313:
2309:
2288:
2285:
2282:
2278:
2257:
2254:
2251:
2247:
2243:
2240:
2237:
2233:
2229:
2226:
2223:
2220:
2217:
2213:
2192:control energy
2173:
2169:
2165:
2161:
2157:
2132:
2110:
2084:
2062:
2037:
2015:
1990:
1985:
1980:
1977:
1972:
1968:
1964:
1960:
1939:
1936:
1933:
1930:
1926:
1921:
1917:
1914:
1911:
1908:
1904:
1899:
1895:
1892:
1889:
1886:
1880:
1876:
1846:
1842:
1837:
1834:
1831:
1828:
1824:
1819:
1815:
1812:
1809:
1803:
1797:
1792:
1789:
1786:
1783:
1779:
1774:
1770:
1767:
1764:
1758:
1752:
1747:
1742:
1737:
1733:
1726:
1723:
1717:
1714:
1690:
1687:
1682:
1678:
1652:
1630:
1608:
1586:
1557:
1552:
1547:
1544:
1539:
1535:
1531:
1527:
1506:
1503:
1500:
1497:
1493:
1489:
1486:
1483:
1479:
1475:
1472:
1469:
1466:
1462:
1458:
1455:
1452:
1448:
1444:
1441:
1438:
1435:
1429:
1425:
1395:
1391:
1386:
1383:
1380:
1377:
1373:
1369:
1366:
1363:
1359:
1355:
1352:
1349:
1343:
1337:
1332:
1329:
1326:
1323:
1319:
1315:
1312:
1309:
1305:
1301:
1298:
1295:
1289:
1283:
1277:
1270:
1266:
1258:
1254:
1249:
1242:
1239:
1233:
1230:
1225:
1221:
1217:
1213:
1207:
1202:
1197:
1192:
1188:
1184:
1178:
1172:
1164:
1161:
1155:
1152:
1129:
1126:
1109:
1104:
1099:
1095:
1091:
1086:
1081:
1077:
1073:
1070:
1067:
1064:
1059:
1047:
1044:
1041:
1038:
1033:
1021:
990:
970:
955:endpoint cost
942:
922:
900:
896:
873:
869:
848:
824:
821:
818:
787:
784:
781:
754:
751:
748:
743:
739:
735:
732:
727:
723:
719:
709:
704:
700:
696:
693:
688:
684:
680:
670:
640:
630:
627:
624:
621:
618:
615:
612:
602:
599:
596:
593:
583:
563:the algebraic
552:
549:
546:
543:
540:
537:
534:
524:
521:
518:
515:
504:
493:
490:
487:
484:
478:
459:state equation
446:
442:
437:
434:
431:
428:
425:
422:
412:
409:
406:
403:
393:
389:
382:
378:
370:
366:
361:
357:
354:
349:
345:
341:
338:
333:
329:
325:
315:
310:
306:
302:
299:
294:
290:
286:
276:
272:
269:
266:
261:
257:
253:
248:
244:
240:
237:
234:
231:
221:
218:
215:
212:
202:
199:
110:
109:General method
107:
103:control theory
87:Lev Pontryagin
32:control theory
15:
9:
6:
4:
3:
2:
6858:
6847:
6844:
6842:
6839:
6838:
6836:
6827:
6823:
6821:
6818:
6816:
6813:
6810:
6807:
6804:
6800:
6798:
6794:
6791:
6788:
6786:
6783:
6781:
6778:
6776:
6773:
6772:
6771:
6767:
6764:
6762:
6759:
6757:
6754:
6751:
6747:
6745:
6742:
6731:
6730:
6725:
6720:
6719:
6709:
6707:0-13-638098-0
6703:
6699:
6698:
6693:
6689:
6685:
6683:0-444-01609-0
6679:
6675:
6674:
6669:
6665:
6664:Kamien, M. I.
6661:
6657:
6655:0-387-90155-8
6651:
6647:
6646:
6641:
6640:Rishel, R. W.
6637:
6633:
6629:
6627:0-470-11481-9
6623:
6619:
6618:
6613:
6609:
6608:Bryson, A. E.
6605:
6601:
6599:1-886529-11-6
6595:
6591:
6587:
6583:
6582:
6570:
6566:
6562:
6558:
6553:
6548:
6544:
6540:
6536:
6532:
6528:
6524:
6520:
6513:
6505:
6501:
6497:
6493:
6489:
6485:
6481:
6477:
6473:
6469:
6465:
6461:
6457:
6450:
6443:
6437:
6430:
6424:
6418:
6412:
6405:
6404:
6398:
6394:
6391:
6386:
6379:
6375:
6370:
6363:
6357:
6351:
6348:
6345:Ross, I. M.,
6342:
6334:
6330:
6326:
6320:
6316:
6313:
6308:
6298:
6288:
6278:
6270:
6266:
6262:
6258:
6254:
6250:
6243:
6236:
6230:
6223:
6217:
6210:
6204:
6196:
6190:
6186:
6179:
6171:
6167:
6163:
6159:
6155:
6151:
6145:
6136:
6129:
6123:
6115:
6109:
6105:
6101:
6095:
6087:
6083:
6079:
6075:
6071:
6070:Bryson, A. E.
6065:
6056:
6051:
6047:
6043:
6039:
6035:
6031:
6027:
6021:
6013:
6009:
6004:
5999:
5995:
5991:
5987:
5980:
5971:
5966:
5959:
5951:
5947:
5943:
5937:
5933:
5932:
5924:
5916:
5914:0-471-02594-1
5910:
5906:
5901:
5900:
5894:
5888:
5880:
5876:
5872:
5866:
5862:
5855:
5853:
5851:
5849:
5844:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5777:Kalman filter
5775:
5772:
5771:JModelica.org
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5713:
5689:
5685:
5677:
5672:
5668:
5665:
5662:
5659:
5655:
5648:
5643:
5639:
5636:
5633:
5630:
5627:
5624:
5621:
5617:
5610:
5604:
5598:
5591:
5571:
5564:
5558:
5555:
5552:
5543:
5537:
5534:
5532:
5524:
5518:
5509:
5503:
5492:
5486:
5483:
5480:
5471:
5468:
5466:
5458:
5449:
5446:
5417:
5411:
5388:
5382:
5373:
5360:
5357:
5351:
5345:
5325:
5316:
5297:
5292:
5283:
5277:
5269:
5263:
5257:
5252:
5249:
5243:
5235:
5226:
5223:
5221:
5213:
5204:
5201:
5191:
5188:
5179:
5173:
5165:
5159:
5153:
5150:
5144:
5138:
5135:
5132:
5129:
5127:
5119:
5111:
5095:
5089:
5083:
5077:
5074:
5065:
5059:
5052:
5044:
5038:
5032:
5026:
5020:
5017:
5014:
5012:
5007:
4994:
4978:
4972:
4969:
4966:
4960:
4951:
4948:
4922:
4916:
4896:
4893:
4889:
4879:
4873:
4866:
4858:
4852:
4846:
4840:
4834:
4831:
4827:
4821:
4816:
4812:
4808:
4775:
4758:
4754:
4731:
4727:
4718:
4699:
4693:
4690:
4687:
4683:
4679:
4676:
4673:
4670:
4662:
4658:
4654:
4652:
4645:
4642:
4639:
4635:
4625:
4620:
4615:
4609:
4606:
4603:
4599:
4595:
4592:
4588:
4581:
4576:
4573:
4570:
4566:
4562:
4560:
4553:
4549:
4522:
4518:
4495:
4491:
4481:
4468:
4465:
4460:
4456:
4435:
4426:
4407:
4402:
4395:
4391:
4385:
4381:
4375:
4370:
4367:
4359:
4355:
4346:
4337:
4334:
4332:
4325:
4321:
4317:
4312:
4309:
4306:
4302:
4294:
4291:
4284:
4280:
4274:
4270:
4264:
4261:
4256:
4253:
4250:
4246:
4242:
4239:
4236:
4234:
4224:
4220:
4211:
4196:
4192:
4186:
4183:
4180:
4176:
4172:
4165:
4161:
4155:
4150:
4146:
4140:
4135:
4131:
4127:
4124:
4122:
4117:
4104:
4089:
4085:
4081:
4078:
4073:
4069:
4065:
4060:
4057:
4054:
4050:
4028:
4024:
4002:
3994:
3990:
3984:
3979:
3975:
3969:
3964:
3960:
3956:
3952:
3946:
3943:
3940:
3935:
3932:
3929:
3925:
3921:
3888:
3887:
3885:
3868:
3862:
3842:
3822:
3799:
3793:
3789:
3783:
3775:
3769:
3746:
3740:
3717:
3711:
3689:
3685:
3664:
3644:
3624:
3609:
3606:
3598:
3588:
3587:the talk page
3584:
3578:
3574:
3572:
3567:This section
3565:
3556:
3555:
3547:
3530:
3524:
3501:
3495:
3486:
3468:
3462:
3439:
3433:
3425:
3414:
3411:
3403:
3393:
3389:
3385:
3379:
3378:
3374:
3369:This section
3367:
3363:
3358:
3357:
3349:
3347:
3346:
3340:
3336:
3335:discrete time
3332:
3328:
3318:
3316:
3312:
3308:
3307:
3302:
3298:
3294:
3293:
3288:
3284:
3280:
3276:
3271:
3267:
3263:
3259:
3255:
3251:
3246:
3224:
3222:
3192:
3190:
3137:
3128:
3126:
3125:cost function
3122:
3117:
3093:
3089:
3052:
3028:
3025:
3022:
3019:
2980:
2971:
2968:
2966:
2958:
2932:
2923:
2921:
2913:
2893:
2889:
2885:
2881:
2877:
2862:
2860:
2856:
2852:
2759:
2741:
2717:
2714:
2694:
2672:
2659:
2656:
2643:
2638:
2623:
2613:
2605:
2601:
2583:
2565:
2559:
2532:
2529:
2511:
2500:
2494:
2469:
2458:
2447:
2444:
2438:
2429:
2413:
2394:
2366:
2360:
2333:
2330:
2320:
2314:
2283:
2252:
2238:
2227:
2224:
2218:
2201:
2195:
2193:
2189:
2188:
2163:
2147:
2099:
2052:
2003:
1988:
1978:
1970:
1966:
1937:
1931:
1915:
1909:
1893:
1887:
1878:
1862:
1857:
1844:
1829:
1810:
1790:
1784:
1765:
1735:
1731:
1724:
1721:
1715:
1712:
1704:
1680:
1676:
1667:
1575:
1570:
1555:
1545:
1537:
1533:
1504:
1498:
1484:
1473:
1467:
1453:
1442:
1436:
1427:
1411:
1406:
1393:
1378:
1364:
1350:
1330:
1324:
1310:
1296:
1268:
1264:
1256:
1252:
1247:
1240:
1237:
1231:
1223:
1219:
1205:
1190:
1186:
1162:
1159:
1153:
1150:
1142:
1138:
1136:
1125:
1123:
1102:
1097:
1093:
1089:
1084:
1079:
1075:
1071:
1065:
1057:
1045:
1039:
1031:
1010:
1006:
1005:
988:
968:
960:
956:
940:
920:
898:
894:
871:
867:
846:
838:
819:
801:
782:
752:
749:
741:
737:
733:
725:
721:
707:
702:
698:
694:
686:
682:
654:
638:
628:
622:
619:
613:
600:
594:
566:
550:
544:
541:
535:
522:
516:
491:
485:
476:
460:
444:
432:
429:
423:
410:
404:
387:
380:
376:
368:
364:
359:
355:
347:
343:
339:
331:
327:
313:
308:
304:
300:
292:
288:
270:
267:
259:
255:
251:
246:
242:
238:
232:
219:
213:
197:
188:
184:
182:
178:
173:
171:
167:
163:
159:
155:
150:
148:
144:
140:
136:
132:
128:
124:
120:
116:
106:
104:
100:
96:
92:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
21:
6811:by Yu-Chi Ho
6770:
6733:. Retrieved
6729:Scholarpedia
6727:
6696:
6672:
6644:
6616:
6589:
6526:
6522:
6512:
6463:
6459:
6449:
6441:
6436:
6423:
6416:
6411:
6400:
6385:
6377:
6369:
6361:
6356:
6346:
6341:
6324:
6307:
6297:
6287:
6277:
6252:
6248:
6242:
6234:
6229:
6221:
6216:
6208:
6203:
6184:
6178:
6161:
6157:
6144:
6135:
6127:
6122:
6103:
6094:
6080:(3): 26–33.
6077:
6073:
6064:
6037:
6033:
6020:
5993:
5989:
5979:
5958:
5930:
5923:
5898:
5887:
5860:
5374:
5317:
4995:
4777:
4482:
4427:
4105:
3890:
3616:
3601:
3595:October 2018
3592:
3581:Please help
3568:
3487:
3424:shadow price
3421:
3406:
3397:
3382:Please help
3370:
3343:
3324:
3304:
3300:
3296:
3290:
3286:
3274:
3269:
3261:
3247:
3129:
3124:
3120:
3118:
3091:
3087:
2894:of the form
2875:
2868:
2854:
2850:
2760:
2641:
2639:
2584:
2199:
2196:
2191:
2187:controllable
2185:
2145:
2097:
2004:
1860:
1858:
1702:
1665:
1573:
1571:
1409:
1407:
1140:
1134:
1131:
1121:
1008:
1002:
959:running cost
958:
954:
836:
799:
564:
458:
189:
185:
174:
165:
161:
157:
153:
151:
129:is a set of
126:
112:
76:
60:unemployment
27:
26:
6735:31 December
6692:Kirk, D. E.
6552:10945/57005
6427:I.M. Ross,
6321:, based on
6150:Ross, I. M.
6100:Ross, I. M.
3550:Finite time
2884:Hamiltonian
181:Constraints
170:constraints
158:control law
6835:Categories
6440:E. Polak,
5970:2005.03186
5840:References
5741:DNSS point
3657:. At date
3573:to readers
3400:April 2018
1009:inequality
1004:Lagrangian
121:that is a
48:spacecraft
6612:Ho, Y.-C.
6561:0731-5090
6488:0077-8923
6012:2405-8963
5950:869522905
5879:625106088
5625:−
5559:λ
5556:−
5487:λ
5484:−
5472:−
5450:˙
5447:λ
5412:λ
5346:λ
5253:−
5241:∂
5233:∂
5227:−
5205:˙
5202:λ
5151:−
5139:λ
5136:−
5117:∂
5109:∂
5078:λ
5075:−
5033:−
4970:−
4952:˙
4847:−
4813:∫
4806:Π
4786:Π
4684:λ
4674:−
4600:λ
4596:−
4567:λ
4550:λ
4519:λ
4457:λ
4371:−
4352:∂
4344:∂
4338:−
4322:λ
4318:−
4303:λ
4262:−
4247:λ
4243:−
4217:∂
4209:∂
4177:λ
4173:−
4141:−
4082:−
4066:−
3970:−
3944:−
3926:∑
3919:Π
3899:Π
3677:there is
3577:evolution
3525:λ
3496:λ
3463:λ
3434:λ
3371:does not
3331:digitally
3225:≤
3130:Minimize
3103:λ
3058:μ
3053:−
3034:λ
2986:∂
2978:∂
2972:−
2959:˙
2956:λ
2942:λ
2938:∂
2930:∂
2914:˙
2742:−
2715:−
2673:−
2660:−
2642:algebraic
2566:−
2530:−
2470:−
2448:−
2430:˙
2331:−
2228:−
1879:˙
1741:∞
1732:∫
1689:∞
1686:→
1428:˙
1248:∫
1141:quadratic
1103:∗
1085:∗
1058:∗
1032:∗
629:≤
477:˙
360:∫
233:⋅
214:⋅
6793:Archived
6694:(1970).
6670:(1991).
6642:(1975).
6614:(1975).
6588:(1995).
6496:16510411
6393:Archived
6374:FALCON.m
6333:35140322
6315:Archived
6102:(2009).
6028:(2000).
5761:GPOPS-II
5709:See also
3637:to date
3352:Examples
2851:constant
2849:are all
2098:constant
2051:matrices
1666:constant
957:and the
651:and the
154:minimize
123:function
6531:Bibcode
6504:7625851
6468:Bibcode
6257:Bibcode
6042:Bibcode
4537:series
3569:may be
3392:removed
3377:sources
3315:FORTRAN
3086:is the
2146:bounded
837:control
835:is the
798:is the
56:economy
36:control
6704:
6680:
6652:
6624:
6596:
6569:756939
6567:
6559:
6502:
6494:
6486:
6331:
6191:
6110:
6010:
5948:
5938:
5911:
5907:–435.
5877:
5867:
5766:CasADi
3311:TOMLAB
3301:GPOPS,
3297:DIRECT
3283:MATLAB
3275:DIRCOL
3262:sparse
3012:where
2827:, and
2268:where
1664:) are
1642:, and
1410:linear
765:where
162:system
64:fiscal
38:for a
6565:S2CID
6500:S2CID
6390:GPOPS
6312:RIOTS
6302:2012.
5965:arXiv
5823:SNOPT
5814:games
3345:RIOTS
3306:PROPT
3287:RIOTS
3279:ASTOS
3266:SNOPT
2890:is a
2200:after
800:state
6737:2022
6702:ISBN
6678:ISBN
6650:ISBN
6622:ISBN
6594:ISBN
6557:ISSN
6492:PMID
6484:ISSN
6464:1065
6329:OCLC
6292:2002
6282:2001
6189:ISBN
6108:ISBN
6008:ISSN
5946:OCLC
5936:ISBN
5909:ISBN
5875:OCLC
5865:ISBN
5736:DIDO
5404:and
4510:and
3375:any
3373:cite
3292:DIDO
2379:and
2122:and
2074:and
2027:and
981:and
933:and
89:and
66:and
52:Moon
6547:hdl
6539:doi
6476:doi
6265:doi
6166:doi
6082:doi
6050:doi
6038:124
5998:doi
5905:393
3386:by
2184:is
149:).
145:(a
137:(a
101:in
6837::
6726:.
6666:;
6638:;
6610:;
6563:.
6555:.
6545:.
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6263:.
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5847:^
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2861:.
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6739:.
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