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Second-Order Conditions for Infinite-Horizon Semilinear Parabolic Control Problems without Tikhonov Regularization

Eduardo Casas, Nicolai Jork

Abstract

We consider semilinear parabolic optimal control problems subject to Neumann boundary conditions, control constraints, and an infinite time horizon. The control constraints are pointwise in time, but they can be pointwise or integral in the space variable. Crucially, the optimal control problem does not include a Tikhonov regularization in the cost functional, which provides a major difficulty in the extension of the classical finite-horizon theory to infinite-horizon optimal control problems. As a consequence of our findings, we establish a sufficient second-order optimality condition and prove that local optimal states of the finite-horizon problems approximate local optimal states to the infinite-horizon problem as the horizon tends to infinity.

Second-Order Conditions for Infinite-Horizon Semilinear Parabolic Control Problems without Tikhonov Regularization

Abstract

We consider semilinear parabolic optimal control problems subject to Neumann boundary conditions, control constraints, and an infinite time horizon. The control constraints are pointwise in time, but they can be pointwise or integral in the space variable. Crucially, the optimal control problem does not include a Tikhonov regularization in the cost functional, which provides a major difficulty in the extension of the classical finite-horizon theory to infinite-horizon optimal control problems. As a consequence of our findings, we establish a sufficient second-order optimality condition and prove that local optimal states of the finite-horizon problems approximate local optimal states to the infinite-horizon problem as the horizon tends to infinity.
Paper Structure (8 sections, 16 theorems, 108 equations)

This paper contains 8 sections, 16 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Assumptions and preliminary results
  3. First-order necessary optimality conditions
  4. Second order sufficient optimality conditions
  5. Box-constraints
  6. Pointwise $L^2$-constraints
  7. Approximation by finite horizon problems
  8. Estimates for the state and adjoint equations

Key Result

Theorem 2.5

For every $u \in L^2(Q_\omega) \cap L^p(0,\infty;L^2(\omega))$, E1.1 has a unique solution $y_u \in W(0,\infty) \cap L^\infty(Q)$. Moreover there exist constants $M_1$ and $M_2$ independent of $u$, $g$, and $y_0$ such that In addition, there exists a constant $M_{ad}$ such that Finally, if $\{u_k\}_{k = 1}^\infty \subset U_{\rm ad}^j$, converges weakly to $u$ in $L^2(Q_\omega)$ then $y_{u_k} \ri

Theorems & Definitions (32)

  • Definition 2.4
  • Theorem 2.5
  • Proof 1
  • Definition 2.6
  • Corollary 2.7
  • Proof 2
  • Theorem 3.1
  • Proof 3
  • Proposition 3.2
  • Proof 4
  • ...and 22 more