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Necessary Conditions In Infinite-Horizon Control Problem That Need No Asymptotic Assumptions

Dmitry V. Khlopin

Abstract

We consider an infinite-horizon optimal control problem with an asymptotic terminal constraint. For the the weakly overtaking criterion and the overtaking criterion, necessary boundary conditions on co-state arcs are deduced, these conditions need no assumptions about the asymptotic behavior of the motion, co-state arc, cost functional, and its derivatives. In the absence of an asymptotic terminal constraint, these boundary conditions with the Pontryagin Maximum Principle allow raising the co-state arcs, corresponding to some asymptotic subdifferentials of the cost functional (fixing the optimal control) at infinity. If this set is a singleton, these conditions coincide with the co-state arc representation proposed by Aseev and Kryazhimskii. These results are illustrated by several examples.

Necessary Conditions In Infinite-Horizon Control Problem That Need No Asymptotic Assumptions

Abstract

We consider an infinite-horizon optimal control problem with an asymptotic terminal constraint. For the the weakly overtaking criterion and the overtaking criterion, necessary boundary conditions on co-state arcs are deduced, these conditions need no assumptions about the asymptotic behavior of the motion, co-state arc, cost functional, and its derivatives. In the absence of an asymptotic terminal constraint, these boundary conditions with the Pontryagin Maximum Principle allow raising the co-state arcs, corresponding to some asymptotic subdifferentials of the cost functional (fixing the optimal control) at infinity. If this set is a singleton, these conditions coincide with the co-state arc representation proposed by Aseev and Kryazhimskii. These results are illustrated by several examples.

Paper Structure

This paper contains 7 sections, 12 theorems, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The statement of the infinite-horizon control problem
  3. Some definitions from set-valued and variational analyses
  4. The main results
  5. Examples
  6. Subdiffentials of lower and upper limits of scalar functions
  7. Proofs of Theorems \ref{['9']} and \ref{['8']}

Key Result

Theorem 1

Under conditions $(H0)$--$(H4)$ and $(E_{\sup})$ let an admissible control process $({\hat{y}},{\hat{u}})$ be locally weakly overtaking optimal for problem sys0_--sysK_. Then, there exists a nonzero solution $({\hat{\psi}},{\hat{\lambda}})\in C(\mathbb{R}_+,\mathbb{X}^*)\times\{0,1\}$ of the corresp

Figures (4)

  • Figure 1: The typical phase diagrams for solutions $(y(\cdot),p(\cdot))$ to \ref{['ds1']} if (a) $\varrho> -1/2$; (b) $\varrho\leq -1/2$.
  • Figure 2: The phase diagram for solutions $(y(\cdot),p(\cdot))$ to \ref{['ds2']} in the case $\varrho=0$.
  • Figure 3: The typical phase diagrams for solutions $(y(\cdot),p(\cdot))$ to \ref{['ds2']} if (a) $\varrho\in(-2;0)$; (b) $\varrho\leq -2$.
  • Figure 4: The typical phase diagrams for solutions $(y(\cdot),p(\cdot))$ of \ref{['ds2']} if (a) $\varrho\in(0;2)$; (b) $\varrho\geq 2$.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • Corollary 2
  • ...and 14 more