In this paper we provide a computational approach to a minimization problem for the value function associated with an affine optimal control problem subject to terminal-constraint with quadratic cost plus a potential, for a fixed final time and initial point. We study the global minimization problem of the value function over the attainable set and the regularity properties of the value function at a global minimizer point. On the other hand, in the global minimization of the value function, by an example in this computational approach, we also focus on a numerical method by Riccati matrix differential equations.
Cite this paper
Zhu, J. (2023). On Global Minimization for the Value Function in Affine Optimal Control Problems. Open Access Library Journal, 10, e408. doi: http://dx.doi.org/10.4236/oalib.1110408.
Barilari, D. and Boarotto, F. (2018) On the Set of Points of Smoothness for the Value Function of Affine Optimal Control Problems. SIAM Journal on Control and Optimization, 56, 649-671. https://doi.org/10.1137/17M1123948
Bardi, M. and Capuzzo-Dolcetta, I. (1997) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston.
https://doi.org/10.1007/978-0-8176-4755-1
Crandall, M.G. and Lions, P.L. (1983) Viscosity Solution of Hamilton-Jacobi Equations. Transactions of the American Mathematical Society, 277, 1-42.
https://doi.org/10.1090/S0002-9947-1983-0690039-8
Fleming, W.H. (1969) The Cauchy Problem for a Nonlinear First Order Partial Differential Equation. Journal of Differential Equations, 5, 515-530.
https://doi.org/10.1016/0022-0396(69)90091-6
Agrachev, A.A., Barilari, D. and Boscain, U. (2012) Introduction to Riemannian and Sub-Riemannian Geometry.
https://people.sissa.it/~agrachev/agrachev_files/ABB-final-SRnotes.pdf
Trelat, E. (2000) Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost. Journal of Dynamical and Control Systems, 61, 511-541.
Sussmann, H.J. and Jurdjevic, V. (1972) Controllability of Nonlinear Systems. Journal of Differential Equations, 12, 95-116.
https://doi.org/10.1016/0022-0396(72)90007-1
Zhu, J.H. (2005) On Stochastic Riccati Equations for the Stochastic LQR Problem. System and Control Letters, 54, 119-124.
https://doi.org/10.1016/j.sysconle.2004.07.003