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Generic Solutions to Controlled Balance Laws

Alberto Bressan, Khai T. Nguyen

Abstract

The paper is concerned with a scalar balance law, where the source term depends on a control function $α(t)$. Given a control $α\in \mathbf{L}^\infty\bigl([0,T]\bigr)$, it is proved that, for generic initial data $\bar u \in \mathcal{C}^3(\mathbb{R})$, the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time $T$ no shock interaction occurs, and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time $T$. Such behavior persists under any suitably small perturbation of the flux, source, and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.

Generic Solutions to Controlled Balance Laws

Abstract

The paper is concerned with a scalar balance law, where the source term depends on a control function . Given a control , it is proved that, for generic initial data , the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time no shock interaction occurs, and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time . Such behavior persists under any suitably small perturbation of the flux, source, and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.

Paper Structure

This paper contains 4 sections, 4 theorems, 131 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:1']}
  3. Proof of Theorem \ref{['t:2']}
  4. Generic behavior of optimally controlled balance laws

Key Result

Theorem 1.1

Let $T>0$ and let $f,g,\alpha$ be given, satisfying (A1). Then there is a ${\cal G}_\delta$ set of initial data ${\cal M}\subset{\cal C}^3(I\!\!R)$ with the following property. For every initial data $\bar{u}\in {\cal M}$, the solution to (cblaw) is piecewise continuous, containing finitely many sh

Figures (7)

  • Figure 1: Left: a generic, structurally stable shock configuration. Right: examples of non-generic configurations. These occur because: (i) Three shocks interact simultaneously. (ii) Two shocks interact exactly at the terminal time $t=T$. (iii) Along the characteristic starting at $\bar{x}$, the gradient blows up exactly at the time when another shock is reached. (iv) Along the characteristic starting at $\bar{y}$, the gradient blows up exactly at the terminal time $t=T$. These four features are all structurally unstable, since they no longer hold for an arbitrarily small perturbation of the initial data.
  • Figure 2: By the relative position of the curves implicitly defined by $\theta=0$ and $\theta_y=0$, one concludes that the local minimum of the function $T(\cdot)$ is isolated.
  • Figure 3: Four non-generic configurations. In each case, the solution satisfies a system where the number of equations is strictly larger than the number of variables.
  • Figure 4: The singularities at the points $P_1,\ldots,P4$ are non generic. According to Definition \ref{['def:31']}, their indices are: $N(P_1) = 2$, $N(P_2)=1$, $N(P_3)=2$, $N(P_4)=1$. On the other hand, the singularities at $P_5,P_6$ are generic, hence $N(P_5)= N(P_6)=0$. Here the dotted lines indicate characteristics where the gradient $u_x$ blows up at the terminal point.
  • Figure 5: Perturbing the initial data to avoid a triple shock interaction at $P$. Here the initial data $\bar{u}$ is slightly increased in a neighborhood of $y_1$, as in (\ref{['pert3']}). This perturbation produces a forward shift of the shock at $x_1(t)$. As a result, the three incoming shocks no longer interact all at the same point.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1