Global minima for semilinear optimal control problems

Abstract

We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. Numerical examples with unique global solutions are presented.

Appendix

Lemma 7.1

We have for $a,b \ge 0, \lambda ,\mu >0$ that

$$\begin{aligned} a^{\lambda } b^{\mu } \le \frac{\lambda ^{\lambda } \mu ^{\mu }}{(\lambda +\mu )^{\lambda +\mu }} (a+b)^{\lambda +\mu }. \end{aligned}$$

Proof

Apply Young’s inequality $xy \le \tfrac{1}{P} \, x^P + \tfrac{1}{Q} \, y^Q \, x,y \ge 0, \, \tfrac{1}{P}+\tfrac{1}{Q}=1$ to $P=\frac{\lambda + \mu }{\lambda }$, $Q=\frac{\lambda + \mu }{\mu }$ and $x= \left( P a \right) ^{\frac{1}{P}}, \, y= \left( Q b \right) ^{\frac{1}{Q}}$. $\square$

Lemma 7.2

Suppose that Assumption 1 holds. Then we have for $a,b \in {\mathbb {R}}$

$$\begin{aligned} \left| \int _0^1 \phi '\left( t a + (1-t)b \right) -\phi '(b) \, dt \right| \le | a - b| L_r \left( \int _0^1 \phi '\left( t a+ (1-t)b \right) \, dt \right) ^{\frac{1}{r}}, \end{aligned}$$

where

$$\begin{aligned} L_r{:}= M \left( \frac{r-1}{2r-1} \right) ^{\frac{r-1}{r}}. \end{aligned}$$

Proof

We start by noticing that

$$\begin{aligned} \int _0^1 \phi '\left( t a+ (1-t)b \right) -\phi '(b) \, dt&= \int _0^1 \int _0^t \phi ''\left( \tau a+ (1-\tau )b \right) (a-b) \, d\tau \, dt \\&= (a-b) \int _0^1 (1-t)\phi ''\left( t a+ (1-t)b \right) \, dt. \end{aligned}$$

Therefore, taking the absolute value and using Assumption 1 we get

$$\begin{aligned} \left| \int _0^1 \phi '\left( t a+ (1-t)b \right) -\phi '(b) \, dt \right|\le & {} | a - b | M \int _0^1 (1-t)\phi '\left( t a+ (1-t)b \right) ^{\frac{1}{r}} \, dt \\\le & {} | a - b | M \Vert 1-t \Vert _{L^{r'}(0,1)}\nonumber \\&\times \, \left( \int _0^1 \phi '\left( t a+ (1-t)b \right) \, dt \right) ^{\frac{1}{r}}, \end{aligned}$$

where $\tfrac{1}{r}+\tfrac{1}{r'}=1$. It is easy to see that

$$\begin{aligned} \Vert 1-t \Vert _{L^{r'}(0,1)} = \left( \frac{1}{r'+1} \right) ^{\frac{1}{r'}}=\left( \frac{r-1}{2r-1} \right) ^{\frac{r-1}{r}}. \end{aligned}$$

Denoting $M \Vert 1-t \Vert _{L^{r'}(0,1)}$ by $L_r$ completes the proof. $\square$

Theorem 7.3

(Gagliardo–Nirenberg interpolation inequality)

For $2 \le q < \infty$ we define $\mu =1-\frac{2}{q}$ as well as

$$\begin{aligned} GN_q{:}= \sup _{f \in H^1({\mathbb {R}}^2), f \ne 0} \frac{\Vert f \Vert _{L^q({\mathbb {R}}^2)}}{\Vert f \Vert _{L^2({\mathbb {R}}^2)}^{1-\mu } \Vert \nabla f \Vert _{L^2({\mathbb {R}}^2)}^{\mu }}. \end{aligned}$$

Then $GN_q \le C_q{:}= \min (C^{(1)}_q,C^{(2)}_q,C^{(3)}_q)$, where

$$\begin{aligned} C^{(1)}_q= & {} \left( \mu C_{2,2 \mu } \right) ^{-\mu }, \quad \text{ if } q \ge 4; \end{aligned}$$

(7.1)

$$\begin{aligned} C^{(2)}_q= & {} \frac{1}{\sqrt{\mu ^{\mu } (1- \mu )^{1 -\mu }}} \left( 2 \pi B \left( 1,\frac{2(1-\mu )}{2 \mu }\right) \right) ^{\mu /2} k_B \left( \frac{4}{2+2 \mu }\right) ; \end{aligned}$$

(7.2)

$$\begin{aligned} C^{(3)}_q= & {} \left( \frac{1}{\pi } \right) ^{\frac{q-2}{2q}} \prod _{j=2}^{\infty } \left( \frac{2^j}{2^j +q-2} \right) ^{\frac{2^j+2-q}{2^j q}}. \end{aligned}$$

(7.3)

Here,

$$\begin{aligned} C_{2,s}= & {} 2^{1/s} \left( \frac{2-s}{s-1} \right) ^{(s-1)/s} \left( 2 \pi B \left( \frac{2}{s},3-\frac{2}{s}\right) \right) ^{1/2}, \; 1< s < 2; \; C_{2,1}=2 \sqrt{\pi }; \\ B(a,b)= & {} \frac{\Gamma (a) \Gamma (b)}{\Gamma (a+b)}, \quad a,b>0 \\ k_B(p)= & {} \left( \frac{p}{2 \pi } \right) ^{1/p} \left( \frac{p'}{2 \pi } \right) ^{- 1/p'}, \quad \frac{1}{p} + \frac{1}{p'}=1. \end{aligned}$$

Proof

The bounds (7.1) and (7.2) can be found in the paper [19] by Veling. We remark that $GN_q=\lambda _{2,\mu }^{-1}$, where $\lambda _{2,\mu }$ is defined in [19, (1.7)]. The estimate (7.1) is [19, (1.31)] (note that $\mu \ge \frac{1}{2} \Leftrightarrow q \ge 4$), while (7.2) is [19, (1.42), (1.43)], where the latter bound has been proved by Nasibov in [15].

Let us now turn to the proof of (7.3). To begin, we claim that for all $k \in {\mathbb {N}}_0$

$$\begin{aligned} \displaystyle \Vert f \Vert _{L^q} \le \left( \frac{1}{\pi } \right) ^{\frac{1}{2}(1 - \frac{q_k}{q})} \prod _{j=2}^{k+1} \left( \frac{2^j}{2^j +q-2} \right) ^{\frac{2^j+2-q}{2^j q}} \Vert f \Vert _{L^{q_k}}^{\frac{q_k}{q}} \Vert \nabla f \Vert _{L^2}^{1-\frac{q_k}{q}}, \end{aligned}$$

(7.4)

where

$$\begin{aligned} q_k = 2^{-k} \left( q+ 2( 2^k -1) \right) . \end{aligned}$$

The inequality clearly holds for $k=0$. Suppose that (7.4) is true for some $k \in {\mathbb {N}}_0$. We infer from Theorem 1 in [9] for the case $d=2$ that

$$\begin{aligned} \displaystyle \Vert f \Vert _{L^{2p}} \le A \Vert f \Vert _{L^{p+1}}^{1-\theta } \Vert \nabla f \Vert _{L^2}^{\theta }, \qquad 1<p<\infty . \end{aligned}$$

(7.5)

Here,

$$\begin{aligned}&A= \left( \frac{y(p-1)^2}{4 \pi } \right) ^{\frac{\theta }{2}} \left( \frac{2y-2}{2y} \right) ^{\frac{1}{2p}} \left( \frac{\Gamma (y)}{ \Gamma (y-1)} \right) ^{\frac{\theta }{2}} \quad \text{ with } \quad \theta = \frac{2(p-1)}{4p},\\&\quad y= \frac{p+1}{p-1}. \end{aligned}$$

Using the formula for y and observing that $\Gamma (y)= (y-1) \Gamma (y-1)$, the expression for A can be simplified to

$$\begin{aligned} A = \left( \frac{1}{\pi } \right) ^{\frac{\theta }{2}} \left( \frac{p+1}{2} \right) ^{\frac{\theta }{2} - \frac{1}{2p}}. \end{aligned}$$

We apply (7.5) for $p=\frac{1}{2} q_k$ and obtain

$$\begin{aligned} \displaystyle \Vert f \Vert _{L^{q_k}} \le A \Vert f \Vert _{L^{\frac{1}{2} q_k+1}}^{1-\theta } \Vert \nabla f \Vert _{L^2}^{\theta }, \end{aligned}$$

(7.6)

where

$$\begin{aligned} A= \left( \frac{1}{\pi } \right) ^{\frac{\theta }{2}} \left( \frac{\frac{1}{2} q_k+1}{2} \right) ^{\frac{\theta }{2} - \frac{1}{q_k}} \quad \text{ and } \quad \theta = \frac{q_k -2}{2 q_k}. \end{aligned}$$

Since $\frac{1}{2} q_k+1= q_{k+1}$ we find that

$$\begin{aligned} A= \left( \frac{1}{\pi } \right) ^{\frac{\theta }{2}} \left( \frac{q_{k+1}}{2} \right) ^{\frac{\theta }{2} - \frac{1}{q_k}} \quad \text{ and } \quad \theta = 1 - \frac{q_{k+1}}{q_k}, \end{aligned}$$

which, inserted into (7.6) yields

$$\begin{aligned} \displaystyle \Vert f \Vert _{L^{q_k}} \le \left( \frac{1}{\pi } \right) ^{\frac{\theta }{2}} \left( \frac{q_{k+1}}{2} \right) ^{\frac{\theta }{2} - \frac{1}{q_k}} \Vert f \Vert _{L^{q_{k+1}}}^{1-\theta } \Vert \nabla f \Vert ^{\theta }_{L^2}. \end{aligned}$$

(7.7)

Using the induction hypothesis we infer

$$\begin{aligned} \Vert f \Vert _{L^q}\le & {} \left( \frac{1}{\pi } \right) ^{\frac{1}{2}(1 - \frac{q_k}{q})+\frac{\theta }{2} \frac{q_k}{q}} \left( \frac{q_{k+1}}{2} \right) ^{\left( \frac{\theta }{2} - \frac{1}{q_k}\right) \frac{q_k}{q}} \\&\times \,\prod _{j=2}^{k+1} \left( \frac{2^j}{2^j +q-2} \right) ^{\frac{2^j+2-q}{2^j q}} \Vert f \Vert _{L^{q_{k+1}}}^{(1-\theta )\frac{q_k}{q}} \Vert \nabla f \Vert _{L^2}^{1-\frac{q_k}{q}+\theta \frac{q_k}{q}}. \end{aligned}$$

Elementary calculations show that

$$\begin{aligned} \frac{1}{2} \left( 1 - \frac{q_k}{q} \right) +\frac{\theta }{2} \frac{q_k}{q}= & {} \frac{1}{2} \left( 1 - \frac{q_{k+1}}{q} \right) , \\ \left( \frac{q_{k+1}}{2} \right) ^{\left( \frac{\theta }{2} - \frac{1}{q_k}\right) \frac{q_k}{q}}= & {} \left( \frac{2^{k+2}}{2^{k+2} + q -2} \right) ^{\frac{2^{k+2}+2-q}{2^{k+2} q}}, \\ (1-\theta ) \frac{q_k}{q}= & {} \frac{q_{k+1}}{q}, \\ 1-\frac{q_k}{q}+\theta \frac{q_k}{q}= & {} 1 - \frac{q_{k+1}}{q}, \end{aligned}$$

which implies (7.4) for $k+1$. The result now follows by sending $k \rightarrow \infty$ in (7.4) and by observing that $\lim _{k \rightarrow \infty } q_k = 2$. $\square$

Fig. 4

The values of the constants $C^{(2)}_q,C^{(3)}_q$ over the range $2 \le q \le 10$ and $C^{(1)}_q$ over $4 \le q \le 10$

Figure 4 illustrates the values of the constants (7.1)–(7.3) for a certain range of q, namely, $2 \le q \le 10$ for $C_q^{(2)},C_q^{(3)}$ and $4 \le q \le 10$ for $C_q^{(1)}$. We can see clearly that the values of $C_q^{(1)}$ are smaller than those of $C_q^{(2)},C_q^{(3)}$ for approximately $q \ge 6$. In order to derive a computable upper bound on $C_q^{(3)}$ we note that $\frac{2^j}{2^j+q-2} \le 1$ for $j\in {\mathbb {N}}$, and therefore

$$\begin{aligned} C^{(3)}_q \le \left( \frac{1}{\pi } \right) ^{\frac{q-2}{2q}} \prod _{j=2}^{k-1} \left( \frac{2^j}{2^j +q-2} \right) ^{\frac{2^j+2-q}{2^j q}}, \ k\ge k_0, \end{aligned}$$

where $k_0\ge 2$ is chosen so large that $2^{k_0}+2-q\ge 0$. In our calculations we used $k-1=200$. All the computations of these constants are done using Mathematica 8.