The value functions of singularly perturbed time-optimal control problems in the framework of Lyapunov functions method

✍ Scribed by N.N. Subbotina

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 247 KB
• Volume: 45
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/j.mcm.2006.11.004

✦ Synopsis

The Dirichlet problems for singularly perturbed Hamilton-Jacobi-Bellman equations are considered. Some impulse variables in the Hamiltonians have coefficients with a small parameter of singularity ε in denominators.

The research appeals to the theory of minimax solutions to HJEs. Namely, for any ε > 0, it is known that the unique lower semicontinuous minimax solution to the Dirichlet problem for HJBE coincides with the value function u ε of a time-optimal control problem for a system with fast and slow motions.

Effective sufficient conditions based on the fact are suggested for functions u ε to converge, as ε tends to zero. The key condition is existence of a Lyapunov type function providing a convergence of singularly perturbed characteristics of HJBEs to the origin. Moreover, the convergence implies equivalence of the limit function u 0 and the value function of an unperturbed time-optimal control problem in the reduced subspace of slow variables.