Approximation of Optimal Control Problems with State Constraints: Estimates and Applications

We present some a priori estimates for the rate of convergence of two approximation schemes related to the deterministic infinite horizon problem with state constraints. A first order and a second order scheme are studied in detail and some hints on the construction of higher order methods are given. We prove that the schemes converge to the constrained viscosity solution of the related Hamilton-Jacobi-Bellman equation and we show that they can also be used to produce approximate optimal trajectories. The above results are applied to the numerical solution of a Vidale-Wolfe advertising model with state constraint.

[1]  M. L. Vidale,et al.  An Operations-Research Study of Sales Response to Advertising , 1957 .

[2]  S. Sethi.  Dynamic Optimal Control Models in Advertising: A Survey , 1977 .

[3]  I. Dolcetta.  On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming , 1983 .

[4]  R. González,et al.  On deterministic control problems: An approximation procedure for the optimal cost , 1983, The 22nd IEEE Conference on Decision and Control.

[5]  H. Ishii,et al.  Approximate solutions of the bellman equation of deterministic control theory , 1984 .

[6]  R. González,et al.  On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost I. The Stationary Problem , 1985 .

[7]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[8]  P. Loreti.  Some properties of constrained viscosity solutions of Hamilton-Jacobi Bellman equations , 1987 .

[9]  M. Falcone.  A numerical approach to the infinite horizon problem of deterministic control theory , 1987 .

[10]  A. Wyner,et al.  Analysis and Optimization of Systems , 1988 .

[11]  M. Falcone,et al.  Discrete Dynamic Programming and Viscosity Solutions of the Bellman Equation , 1989 .

[12]  G. Barles,et al.  Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations , 1990 .

[13]  M. Falcone,et al.  Discrete approximation of the minimal time function for systems with regular optimal trajectories , 1990 .

[14]  P. Lions,et al.  Hamilton-Jacobi equations with state constraints , 1990 .

[15]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[16]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1991 .

[17]  Maurizio Falcone,et al.  Corrigenda: A numerical approach to the infinite horizon problem of deterministic control theory , 1991 .

[18]  Piero Lanucara,et al.  A splitting algorithm for Hamilton-Jacobi-Bellman equations , 1992 .

[19]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[20]  M. Sun.  Domain decomposition algorithms for solving hamilton-jacobi-bellman equations , 1993 .

[21]  J. R. Dorroh,et al.  A Multistate, Multicontrol Problem with Unbounded Controls , 1994 .

[22]  Marizio Falcone,et al.  Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations , 1994 .

[23]  B. Craven.  Control and optimization , 2019, Mathematical Modelling of the Human Cardiovascular System.