Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse control for VCHE with abstract J

Giang Nguyen Hai Ha

Abstract

We investigate a distributed optimal control problem for the viscous Camassa--Holm equations with sparse controls and a general cost functional. Considering three different forms of sparsity-promoting terms, we prove the existence of optimal solutions, derive the corresponding optimality conditions and analyze the stability of optimal solutions with respect to the sparsity parameter.

Sparse control for VCHE with abstract J

Abstract

We investigate a distributed optimal control problem for the viscous Camassa--Holm equations with sparse controls and a general cost functional. Considering three different forms of sparsity-promoting terms, we prove the existence of optimal solutions, derive the corresponding optimality conditions and analyze the stability of optimal solutions with respect to the sparsity parameter.

Paper Structure

This paper contains 10 sections, 24 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces and inequalities for the nonlinear terms
  4. Existence and uniqueness of weak solutions to the viscous Camassa - Holm equations
  5. The control-to-state mapping
  6. Assumptions on the objective functional and the set of admissible control
  7. Existence of optimal solutions
  8. First-order necessary optimality conditions
  9. Second-order optimality conditions
  10. Stability of optimal control problem

Key Result

Lemma 2.1

If $n=3$ then the following estimates hold $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (48)

  • Lemma 2.1: temam1995navier,robinson2003
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.2
  • Theorem 2.2
  • Theorem 2.3
  • Example 2.1
  • Example 2.2
  • ...and 38 more