zbMATH — the first resource for mathematics

Singular perturbation of a finite horizon problem with state-space constraints. (English) Zbl 0953.49031
The paper deals with optimal control problems for systems containing fast and slow subsystems under geometrical constraints in the state subspace of fast variables. Cost functionals of the Bolza type are considered.
Asymptotics to the singularly perturbed control problems are studied as a parameter of singularity tends to zero (it means that velocities of singularly perturbed fast variables tend to infinity).
Sufficient conditions for convergence are obtained in the context and with equipment of constrained viscosity solutions to corresponding Hamilton-Jacobi-Bellman equations. The conditions provide controllability property for fast subsystems and weak invariance property for the geometrical state constraints.
The results are applied to study the case of Lipschitz controls and exit time optimal control problems.

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
93C73 Perturbations in control/observation systems
35F25 Initial value problems for nonlinear first-order PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
49L20 Dynamic programming in optimal control and differential games
Full Text: DOI