Rotational controls and uniqueness of constrained viscosity solutions of Hamilton-Jacobi PDE

TL;DR

This paper introduces a higher order inward pointing condition involving Lie brackets for control systems, enabling the construction of constrained trajectories and proving the continuity and uniqueness of viscosity solutions of Hamilton-Jacobi PDEs under boundary constraints.

Contribution

It develops a novel higher order inward pointing condition and a rotating control strategy to handle boundary constraints in control systems, extending classical results.

Findings

Constructed constrained trajectories with distance proportional to √d when classical conditions fail.

Proved the continuity of the value function up to the boundary.

Established the uniqueness of the constrained viscosity solution of the Bellman equation.

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.