Rotational controls and uniqueness of constrained viscosity solutions of Hamilton-Jacobi PDE
Abstract
The classical inward pointing condition (IPC) for a control system whose state $x$ is constrained in the closure $C:=\bar\Omega$ of an open set $\Omega$ prescribes that at each point of the boundary $x\in \partial \Omega$ the intersection between the dynamics and the interior of the tangent space of $\bar \Omega$ at $x$ is nonempty. Under this hypothesis, for every system trajectory $x(.)$ on a time-interval $[0,T]$, possibly violating the constraint, one can construct a new system trajectory $\hat x(.)$ that satisfies the constraint and whose distance from $x(.)$ is bounded by a quantity proportional to the maximal deviation $d:=\mathrm{dist}(\Omega,x([0,T]))$. When (IPC) is violated, the construction of such a constrained trajectory is not possible in general. However, for a control system of the form $\dot{x}=f_1(x)u_1+f_2(x)u_2$, we prove in this paper that a "higher order" inward pointing condition involving Lie brackets of the dynamics' vector fields allows for a novel construction of a constrained trajectory $\hat x(.)$ whose distance from the reference trajectory $x(.)$ is bounded by a quantity proportional to $\sqrt{d}$. Our method requires a further assumption of non-positiveness of a sort of curvature and is based on the implementation of a suitable "rotating" control strategy. As an application, we establish the continuity up to the boundary of the value function $V$ of a classical optimal control problem, a continuity that allows to regard $V$ as the unique constrained viscosity solution of the corresponding Bellman equation.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.08530
- arXiv:
- arXiv:2110.08530
- Bibcode:
- 2021arXiv211008530C
- Keywords:
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- Mathematics - Optimization and Control;
- 34H05;
- 49L25
- E-Print:
- 25 pages