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VIDEO LIBRARY |
International youth conference "Geometry & Control"
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From Approximate Reachable Sets to Asymptotic Control Theory Aleksey Fedorov, Alexander Ovseevich Institute for Problems in Mechanics RAS, Moscow, Russia |
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Abstract: 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. Definition. The reachable set 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 Analytically speaking this means that for a state vector \begin{equation*} x=T\frac{\partial {H}_\Omega}{\partial p}(p) \end{equation*} with unknown time Following this strategy, we can make a damping of a non-resonant system of linear oscillators in quasi-optimal time. More precisely, Theorem 1. Assume that system of oscillators is non-resonant. Let \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, 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 $$ f\in L_\infty, \quad \frac{\partial f}{\partial x}\in L_\infty, \quad \frac{\partial f}{\partial t}\in L_\infty. $$ Language: English References
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