Hi All
I am doing a nonlinear optimal control design project, typically a practical engineering problem.
It is easy to solve that with direct method. However, I am wondering is there any way to prove that the optimal solution exists ? Further, since the direct method is employed, is there anyway to prove that the numerical solution will converge to that optimal solution?
Could anyone offer me some reference paper about that ?
thanks
Jie
Dear Jie Yao,
I suggest you to see links i hope that are on subject.
https://www.hindawi.com/journals/isrn/2013/950912/
Article Solving nonlinear optimal control problems with state and co...
Best regards
The answer to the first question depends on the type of the nonlinear system, i.e., ODE or PDE, and the order. For example, see
“Slemrod, Marshall. "Existence of optimal controls for control systems governed by nonlinear partial differential equations." Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 1.3-4 (1974): 229-246.”
“Bors, Dorota, Andrzej Skowron, and Stanislaw Walczak. "On Existence of Solutions to Nonlinear Optimal Control Systems." Dynamic Systems and Applications 21.2 (2012): 441.”
“The existence and uniqueness of solutions to differential equations”
Moreover, the answer also depends on the smoothness or non-smoothness solution of the problem. As we know, determination of the optimal feedback law for nonlinear optimal control problems leads to the Hamilton Jacobi-Bellman (HJB) partial differential equations, and in general, the solutions may not be smooth. The question of existence in the class of unique non-smooth solutions (called the viscosity solutions) is studied in
“M. Crandall and P. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Transactions of the American Mathematical Society, vol. 277, no. 1, pp. 1–42, 1983.”
“M. Bardi and I. Dolcetta, Optimal control and viscosity solutions of Hamilton Jacobi-Bellman equations. Springer, 1997.”
To answer the second question, note that to find the optimal solution, one has to solve an HJB equation. This HJB equation is generally difficult or impossible to solve analytically and therefore, in order to approximately solve the HJB partial differential equations, numerical methods such as Approximate dynamic programming (ADP), which is an efficient and forwarded in time RL method, can be used to generate approximate optimal control policy (near-optimal solution). However, in general, HJB partial differential equations can be solved numerically for very low state dimensions, and moreover, numerical methods generally give up the optimality of the solution in favor of reduced computational complexity, and other implementation-related factors. This implies that the optimal numerical solutions are near-optimal, and the guarantees are mainly provided in the sense of ultimately uniformly boundedness of the solution. Moreover, Banach's fixed-point theorem (i.e., Contraction Mapping Theorem) plays as a key tool to provide guarantees for the existence and uniqueness of fixed points (optimal solutions) of nonlinear optimal control problems, and also provides a practical and powerful method to find those solutions. See the below for more details
https://www.math.ucdavis.edu/~hunter/book/ch3.pdf
https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Buchanan.pdf
Dear Jie Yao,
please find the answer in
1.А.Ф.Филиппов, О некоторых вопросах теории оптимального регулирования // Вестник МГУ, сер. матем., мех., астрон., физ., хим., 1959, №2, с. 25–32.
2.L.Cesari, "Optimization: Theory and Applications", Springer, New-York, 1983.
A good example of the application could be found for example, is in
J Optim Theory Appl (2013) 156:469–492 DOI 10.1007/s10957-012-0128-0 Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field Efstathios Bakolas · Panagiotis Tsiotras
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