Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations

Abstract

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory.

Appendix

Proposition A.1

([14, Proposition 6.1]) Let $\varOmega \subset {\mathbb {R}}^n$ be bounded, open and of class $C^l$. Let $s_1, s_2, s_3$ be real number, $0\le s_1\le l$, $0\le s_2\le l-1$, $0\le s_3\le l$. Assume that

• (i) $s_1+s_2+s_3\ge n/2$ if $s_i\ne n/2$ for all $i=1,2,3$ or

• (ii) $s_1+s_2+s_3>n/2$ if $s_i=n/2$ for at least one i.

Then there exists a constant depending on $s_1, s_2, s_3, \varOmega$, scale invariant such that

$$\begin{aligned} |\mathbf{b }(u, v, w)|&\le c|\varOmega |^{\frac{s_1+s_2+s_3}{n}-\frac{1}{2}}\Vert u\Vert _{[s_1]}^{1+[s_1]-s_1}\Vert u\Vert _{[s_1]+1}^{s_1-[s_1]} \Vert v\Vert _{[s_2]+1}^{1+[s_2]-s_2}\Vert v\Vert _{[s_2]+2}^{s_2-[s_2]} \nonumber \\&\quad \cdot \Vert w\Vert _{[s_3]}^{1+[s_3]-s_3}\Vert w\Vert _{[s_3]+1}^{s_3-[s_3]} \end{aligned}$$

(61)

for all $u, v, w\in C^\infty ({{\bar{\varOmega }}})^n.$ Here $[s_i]$ denotes the integer part of $s_i$ and $\Vert \cdot \Vert _0$ denotes the norm of $L^2(\varOmega )$.