Classification of stable time-optimal controls on 2-manifolds
Benedetto Piccoli2006, Journal of Mathematical Sciences
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The research focuses on the classification of stable time-optimal control systems specifically on two-dimensional manifolds. It addresses theoretical frameworks for evaluating control strategies and employs various mathematical techniques to ensure optimality and stability in control processes. The findings contribute valuable insights into the design and analysis of control systems in manifold contexts.
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Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C ∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.
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