Abstract
This chapter describes how to find optimal open-loop and closed-loop impulse controls. We begin by defining an impulse control system and proving the existence and uniqueness of its trajectories, (see also [2, 11]). Then we set up the basic problem of open-loop impulse control. This is how to transfer the system from a given initial state to a given target state within given time under a control of minimum variation. A key point in solving the open-loop impulse control problem is the construction of reachability sets for the system. Here we indicate how to construct such sets and study their properties. After that we present some simple model examples. The solution to the optimal impulse control problem problem is given by the Maximum Rule for Impulse Controls, an analogue of Pontryagin’s Maximum Principle for ordinary controls.
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Notes
- 1.
Following notations of [7], we describe an ellipsoid \({\mathscr {E}\left( {q}, {Q}\right) }\) with parameters \(q \in \mathbb {R}^n\), \(Q \in \mathbb {R}^{n \times n}\), \(Q \ge 0\), as a convex set in \(\mathbb {R}^n\) with support function \(\rho = \left\langle p, q \right\rangle + \left\langle p, Qx \right\rangle ^{\frac{1}{2}}\). If Q is nondegenerate, then \({\mathscr {E}\left( {q}, {Q}\right) } = {\left\{ x \in \mathbb {R}^n \;\big |\; \left\langle x -q, Q^{-1}(x - q) \right\rangle \le 1 \right\} }\).
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Kurzhanski, A.B., Daryin, A.N. (2020). The Open-Loop and Closed-Loop Impulse Controls. In: Dynamic Programming for Impulse Feedback and Fast Controls. Lecture Notes in Control and Information Sciences, vol 468. Springer, London. https://doi.org/10.1007/978-1-4471-7437-0_7
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DOI: https://doi.org/10.1007/978-1-4471-7437-0_7
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