Abstract
In these sections, we deal with additional constraints on the solutions to equations of Sects. 7.1 and 7.4, related to control under generalized (higher) impulses . These restrictions are an analogy of state constraints for systems controlled by ordinary impulses of Chap. 5 (see also [1, 7]). Discussing the problem of optimal control under higher impulses and state constraints we describe it first in terms of the theory of distributions [12, 13]. indicating conditions for its solvability. Then, in order to formulate conditions of optimality, we use a reduction of the system to the first-order form under vector measures, as shown in Sect. 7.3.
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Notes
- 1.
Under the properties of our problem \(\max _t\psi (t,y(t)) = \text {ess}\sup _t \psi (t,y(t)) \),
- 2.
The convolution of generalized function \(x(\cdot )\) with \(\zeta ^{k-1}\) allows to reduce a problem that involves high-order distributions \(x(\cdot )\) to an equivalent in terms of ordinary functions, under notations either \(x^\star (\cdot )\) or \(\mathbf{x}(\cdot )\), depending on those used in literature related to the specific problem.
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Kurzhanski, A.B., Daryin, A.N. (2020). State-Constrained Control Under Higher Impulses. 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_8
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