Continuous Dependence with Respect to the Input of Trajectories of Control-Affine Systems

Abstract

We study the continuous dependence on the input of trajectories of control-affine systems belonging to the class C0 (m) of all systems $\Sigma$ of the form $$\Sigma: \ \ \ \ \ \ \dot{x}=f_0(x)+\sum_{i=1}^m u_i(t)f_i(x),$$ where f0 ,. . . fm are continuous vector fields on some open subset of $\mathbb{R}^n$ and the control functions belong to $L^1([0,T],\mathbb{R}^m)$. We give a simple necessary and sufficient condition for a control sequence $\{u^j\}_{j=1}^{\infty}$ to "${\cal T}^0$-converge" to a control $u^\infty$, i.e., to be such that, for every system $\Sigma$ in C0 , the trajectories generated by the uj converge as $j\rightarrow\infty$ to the trajectories generated by $u^\infty$. We also characterize ${\cal T}^k$-convergence (the concept of control convergence that arises when we use, instead of C0 (m), the class Ck (m) of systems $\Sigma$ where the fi are of class Ck) for $k\ge 1$ in the scalar input case, and we explain how the analogous characterization for the multi-input case fails to be true, unless one restricts oneself to the class $C^k_{comm}(m)$ of systems for which the vector fields f0 ,. . . fm commute. As a preliminary, we define a "topology of trajectory convergence" (or "T-convergence") on the set of all time-varying vector fields $\Omega\times I\ni (x,t)\mapsto f(x,t)\in\mathbb{R}^n$, where $\Omega$ is an open subset of $\mathbb{R}^n$ and I is an interval, and we study some of its properties. This enables us to make the definition of ${\cal T}^k$-convergence precise for sequences and, more generally, for nets, by saying that a net $\{u^\alpha\}_{\alpha\in A}$ in $L^1([0,T],\mathbb{R}^m)$ ${\cal T}^k$-converges to a limit $u^\infty$ if for every system $\Sigma$ in C0 (m) the time-varying vector fields $(x,t)\mapsto f_0(x)+\sum_{i=1}^m u_i^\alpha(t)f_i(x)$ ${\cal T}^k$-converge to $(x,t)\mapsto f_0(x)+\sum_{i=1}^m u_i^\infty(t)f_i(x)$.