From Approximate Reachable Sets to Asymptotic Control Theory

The problem of time-optimal steering of an initial state of a dynamical system to a given manifold is typical for the optimal control theory. Optimal trajectory is to be found as the steepest descent in the direction of the gradient of the cost function. The level sets of the cost functions are boundaries of the reachable set of the system in respect to backward time. The direction of the gradient coincides with the normal to boundary of the reachable set.
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Definition. The reachable set $\mathcal{D}(T)$ is the set of ends at time instant $T$ of all admissible trajectories of the system starting at the given manifold at zero time.
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It is remarkable, that for a wide class of linear systems of the form
\begin{equation*} \dot{x}={A}x+{B}u, \quad |u|\leq1, \end{equation*}
where $u$ is a control, reachable set $\mathcal{D}(T)$ equals asymptotically as $T\to\infty$ to the set $T\Omega$, where $\Omega$ is a fixed convex body, (here given manifold is the origin). More than that, the support function ${H}_\Omega$, which defines $\Omega$ uniquely, has an explicit integral representation. Starting from this point, we can design a control using steepest descent in the normal direction to the boundary of approximate reachable sets $T\Omega$.
Analytically speaking this means that for a state vector $x$ we have to solve the following equation
\begin{equation*} x=T\frac{\partial {H}_\Omega}{\partial p}(p) \end{equation*}
with unknown time $T$ and momentum $p=p(x)$. The control we describe takes the form $u(x)=-{\rm sign}\langle{B,p(x)}\rangle$.
Following this strategy, we can make a damping of a non-resonant system of linear oscillators in quasi-optimal time. More precisely,
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Theorem 1. Assume that system of oscillators is non-resonant. Let $T=T(x)$ be the motion time from the initial point $x$ to the equilibrium under our control, and $\tau=\tau(x)$ be the minimum time. Then, as the $x\to\infty$ we have the asymptotic equality
\begin{equation*} \tau(x)/T(x)=1+o(1). \end{equation*}

These general arguments to a great extent are applicable to the problem of damping of a closed string
\begin{equation*} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2}+u\delta, \quad |u|\leq1. \end{equation*}
Here, $x\in[0,2\pi]$ is the angle coordinate on a one-dimensional torus $\mathcal{T}$, $t$ is time, $\delta$ is the Dirac $\delta$-function. Particularly, we obtain the following result
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Theorem 2. It is possible to damp the string by a bounded load applied to a fixed point in finite time, if at the initial state
$$ f\in L_\infty, \quad \frac{\partial f}{\partial x}\in L_\infty, \quad \frac{\partial f}{\partial t}\in L_\infty. $$