On some extension of optimal control theory

Abstract

Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.

Optimal impulse control of a SIR epidemic

“…Then, restarting at this new point, the process runs until the next intervention and so on. Sometimes, such control is called “singular control.” The goal is to minimize the total (expected) accumulated cost, which may be discounted or not . The popular method of attack to such problems is dynamic programming .…”

“…Impulse control of various dynamical systems attracts the attention of many researchers, eg, those in other works 8,9,[11][12][13][14][15][16][17][18][19][20][21][22][23][24] to mention the most relevant and the most recent works. The underlying system can be described in terms of ordinary 8,9,[11][12][13]20,[22][23][24] or stochastic 17,21 differential equations. In other works, [14][15][16]19 along with the given deterministic drift, there are spontaneous (or natural) Markov jumps of the state.…”

“…21 The goal is to minimize the total (expected) accumulated cost, which may be discounted 12,[14][15][16]18,19,21 or not. 8,12,13,17,20,22 The popular method of attack to such problems is dynamic programming. 12,[14][15][16]18,19,21,23 In other works, 11,20,22,24 versions of the Pontryagin maximum principle are used.…”

“…8,12,13,17,20,22 The popular method of attack to such problems is dynamic programming. 12,[14][15][16]18,19,21,23 In other works, 11,20,22,24 versions of the Pontryagin maximum principle are used.…”

Stabilizability in optimization problems with unbounded data

“…in [17], while in case f and l are functions with a maximal u-growth ν(|u|), by choosing ν(x, u) :=ν(|u|), we can recover the extended Lagrangian and dynamics considered in impulsive control (see e.g. [22,20,12]). The assumptions considered in this paper allow for a vast class of dynamics and Lagrangians, including those with a polynomial dependence on…”