Singular Perturbation Methods in Control: Analysis and Design
Singular Perturbation Methods in Control: Analysis and Design
ISNN '08 Proceedings of the 5th international symposium on Neural Networks: Advances in Neural Networks
Mathematics and Computers in Simulation
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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 @e in denominators. The research appeals to the theory of minimax solutions to HJEs. Namely, for any @e0, it is known that the unique lower semi-continuous minimax solution to the Dirichlet problem for HJBE coincides with the value function u^@e 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^@e to converge, as @e 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.