On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation

Abstract

This paper deals with the numerical implementation of a systematic method for solving bi-objective optimal control problems for wave equations. More precisely, we look for Nash and Pareto equilibria which respectively correspond to appropriate noncooperative and cooperative strategies in multi-objective optimal control. The numerical methods described here consist of a combination of the following: finite element techniques for space approximation; finite difference schemes for time discretization; gradient algorithms for the solution of the discrete control problems. The efficiency of the computational methods is illustrated by the results of some numerical experiments.

Appendix: Existence and uniqueness of a solution to (6)

Let us consider the mapping:

$$ (v_{1}, v_{2}) \in \mathcal{U} \mapsto \left( \frac{\partial J_{1}}{\partial v_{1}}(v_{1}, v_{2}), \frac{\partial J_{2}}{\partial v_{2}}(v_{1}, v_{2}) \right) \in \mathcal{U} . $$

(34)

From the linearity of (2) and the fact that J₁ and J₂ are quadratic, it is obvious that there exist a unique bounded linear operator ${\mathscr{A}}\in {\mathscr{L}}(\mathcal {U} ; \mathcal {U})$ and a unique $b \in \mathcal {U}$ such that:

$$ \left( \frac{\partial J_{1}}{\partial v_{1}}(v_{1}, v_{2}), \frac{\partial J_{2}}{\partial v_{2}}(v_{1}, v_{2}) \right) = \mathscr{A}(v_{1}, v_{2}) - b \ \ \ \ \forall (v_{1},v_{2}) \in \mathcal{U}. $$

Let us identify the mapping ${\mathscr{A}}$. For every $(v_{1}, v_{2}) \in \mathcal {U}$, one has:

where the ϕ_i are given by

(35)

and y is the solution to

(36)

Proposition 1

The mapping ${\mathscr{A}}$ is linear, continuous, and self-adjoint. Furthermore, if μ₁ and μ₂ are sufficiently large (depending on Ω and T), ${\mathscr{A}}$ is strongly positive in $\mathcal {U}$, that is, there exists α₀ > 0 such that:

$$ \left( \mathscr{A}(v_{1},v_{2}),(v_{1},v_{2})\right)_{\mathcal{U}} \geq \alpha_{0} \|(v_{1},v_{2})\|_{\mathcal{U}}^{2} \quad \ \ \forall (v_{1},v_{2}) \in \mathcal{U}. $$

(37)

Proof

The argument is adapted from [13], Proposition 4.1.

Let us see that $\mathcal {A}$ is self-adjoint. Indeed, for any $(v_{1}, v_{2}), (w_{1}, w_{2}) \in \mathcal {U}$, one has:

$$ \begin{array}{@{}rcl@{}} \left( \mathscr{A}(v_{1},v_{2}),(w_{1},w_{2})\right)_{\mathcal{U}} = \iint_{\mathcal{O}_{1} \times (0,T)} \left( \mu_{1} v_{1} + \phi_{1} \right)w_{1} dx dt \\ \noalign{} \phantom{\left( \mathscr{A}(v_{1},v_{2}),(w_{1},w_{2})\right)_{\mathcal{U}}} + \iint_{\mathcal{O}_{2} \times (0,T)} \left( \mu_{2} v_{2} + \phi_{2} \right)w_{2} dx dt . \end{array} $$

(38)

If y and z denote the solutions to (36) corresponding to (v₁,v₂) and (w₁,w₂), we have:

(39)

Thus,

$$ \begin{array}{@{}rcl@{}} \left( \mathscr{A}(v_{1},v_{2}),(w_{1},w_{2})\right)_{\mathcal{U}} &=& \sum\limits_{i=1}^{2} \mu_{i} \iint_{\mathcal{O}_{i} \times (0,T)} v_{i} w_{i} dx dt \\ \noalign{} \phantom{\left( \mathscr{A}(v_{1},v_{2}),(w_{1},w_{2})\right)_{\mathcal{U}}} &+& \iint_{(\mathcal{O}_{1,d} \cup \mathcal{O}_{2,d}) \times (0,T)} y z dx dt \\ \noalign{} \phantom{\left( \mathscr{A}(v_{1},v_{2}),(w_{1},w_{2})\right)_{\mathcal{U}}} &=& \Bigl((v_{1},v_{2}),\mathscr{A}(w_{1},w_{2})\Bigr)_{\mathcal{U}} . \end{array} $$

(40)

Therefore, $\mathcal {A}$ is self-adjoint.

For completeness, let us recall the proof of strong positiveness. Thus, note that, for any $(v_{1},v_{2}) \in \mathcal {U}$:

$$ \left( \mathscr{A}(v_{1},v_{2}),(v_{1},v_{2})\right)_{\mathcal{U}} = \sum\limits_{i=1}^{2} \iint_{\mathcal{O}_{i} \times (0,T)} \left( \mu_{i} v_{i} + \phi_{i} \right)v_{i} dx dt . $$

(41)

Let z₁ (resp. z₂) be the solution to (36) with v₂ = 0 (resp. v₁ = 0). Then, from standard integration by parts, we see that:

$$ \iint_{\mathcal{O}_{i} \times (0,T)} \phi_{i} v_{i} dx dt = \iint_{Q} \phi_{i} (z_{i,tt} - {\varDelta} z_{i}) dx dt = \iint_{\mathcal{O}_{i,d} \times (0,T)} y z_{i} dx dt. $$

(42)

Consequently, there exists C₀ depending on Ω and T such that:

$$ | \iint_{\mathcal{O}_{i} \times (0,T)} \phi_{i} v_{i} dx dt | \leq C_{0} \|(v_{1},v_{2})\|_{\mathcal{U}} \|v_{i}\|_{\mathcal{U}_{i}}, \ \ i=1,2. $$

(43)

From (41) and (43), we see that, if $\min \limits (\mu _{1},\mu _{2}) > \sqrt {2} C_{0}$, then

$$ \left( \mathscr{A}(v_{1},v_{2}),(v_{1},v_{2})\right)_{\mathcal{U}} \geq \left[\min(\mu_{1},\mu_{2}) - \sqrt{2} C_{0} \right] \|(v_{1},v_{2})\|^{2}_{\mathcal{U}} \quad \forall (v_{1},v_{2}) \in \mathcal{U}, $$

whence (37) holds. □

Remark 3

From (41) and (42) we also observe that, if $\mathcal {O}_{1,d}$ and $\mathcal {O}_{2,d}$ coincide and we set $\mathcal {O}_{d} = \mathcal {O}_{i,d}$, then

$$ \left( \mathscr{A}(v_{1},v_{2}),(v_{1},v_{2})\right)_{\mathcal{U}} \geq \min(\mu_{1},\mu_{2}) \|(v_{1},v_{2})\|^{2}_{\mathcal{U}} + \iint_{\mathcal{O}_{d} \times (0,T)} |y|^{2} dx dt \\ $$

for all $(v_{1},v_{2}) \in \mathcal {U}$. Therefore, in this case, ${\mathscr{A}}$ is always strongly positive, regardless of the sizes of the μ_i.

Let us identify b, that is, the nonhomogeneous term in the affine mapping (34). One has:

where $\bar {\phi }_{i}$ is the solution to

(44)

and $\bar {y}$ is the solution of

$$ \left\{ \begin{array} [c]{ll} \bar{y}_{tt} - {\varDelta} \bar{y} = f \ \ & \ \ \text{in} \ Q, \\ \bar{y} = 0 \ \ \ \ & \ \text{on} \ {\varSigma}_{1}, \\ \frac{\partial \bar{y}}{\partial n} = 0 \ & \ \text{on} \ {\varSigma}_{2}, \\ \bar{y}(x,0) = y_{0}(x), \ \ \ \ \ \bar{y}_{t}(x,0) = y_{1}(x) \ \ & \ \ \text{in} \ {\varOmega} . \end{array} \right. $$

(45)

Now, if we define by

$$ a(v, w) := \left( \mathscr{A}v , w \right)_{\mathcal{U}} \quad \ \ \forall v, w \in \mathcal{U} $$

and by

$$ L(w) := \left( b, w\right)_{\mathcal{U}} \ \quad \ \forall w \in \mathcal{U}, $$

we deduce that (6) is equivalent to

$$ a(v, w) = L(w) \quad \forall v \in \mathcal{U}. $$

(46)

From Proposition 1, we see that a(⋅,⋅) is bilinear, continuous, and symmetric. Moreover, if μ₁ and μ₂ are large enough, it is also $\mathcal {U}$-elliptic. On the other hand, L is linear and continuous. Hence, from Lax-Milgram Theorem, the existence and uniqueness of a solution to (46) is ensured if the μ_i are sufficiently large.

Remark 4

Let us fix λ ∈ (0, 1) and let us consider the system (18)–(20). We can adapt the previous argument and prove that there exist other ${\mathscr{A}}$ and b such that (21) holds. Now, we get:

$$ \begin{array}{@{}rcl@{}} \left( \mathscr{A}(v_{1}, v_{2}),(v_{1},v_{2}) \right)_{\mathcal{U}} &=& \iint_{\mathcal{O}_{1}\times(0,T)}\left( \lambda \mu_{1}v_{1} + \lambda \phi_{1} + (1-\lambda)\phi_{2})\right)v_{1} dx dt \\ \noalign{} \phantom{\left( \mathscr{A}(v_{1}, v_{2}),(v_{1},v_{2}) \right)_{\mathcal{U}}} &+& \iint_{\mathcal{O}_{2}\times(0,T)}\left( (1 - \lambda) \mu_{2}v_{2} + \lambda \phi_{1} + (1-\lambda)\phi_{2}\right)v_{2} dx dt \\ \noalign{} \phantom{\left( \mathscr{A}(v_{1}, v_{2}),(v_{1},v_{2}) \right)_{\mathcal{U}}} &\geq& \left( {\min\left( \lambda \mu_{1}, (1-\lambda)\mu_{2} \right)} - \sqrt{2}C_{1}\right)||(v_{1}, v_{2})||^{2}_{\mathcal{U}} \ , \end{array} $$

where C₁ only depends on Ω and T. Thus, if $\min \limits (\mu _{1},\mu _{2}) > \displaystyle \frac {\sqrt {2}C_{1}}{\min \limits (\lambda , 1 - \lambda )}$, the system (18)–(20) possesses exactly one solution.

1.1 Existence and uniqueness of a solution to the semilinear system (23)–(25)

In this section, we consider the semilinear state (i). As before, we assume that is globally Lipschitz-continuous. Thus, there exists C_F > 0 such that:

Note that the couple (v₁,v₂) solves (23)–(25) if and only if it is a fixed point of the nonlinear mapping ${\varLambda }: \mathcal {U} \mapsto \mathcal {U}$, where

$$ \left\{ \begin{array}{l} \displaystyle {\varLambda}(v) = \left( {\varLambda}_{1}(v),{\varLambda}_{2}(v)\right), \ \ {\varLambda}_{i}(v) = -\frac{1}{\mu_{i}} \phi_{i} |_{\mathcal{O}_{i} \times (0,T)} , \\ \phi_{i} \text{is the solution to~(24) for} i=1,2, \\ y \ \text{is} \ \text{the} \ \text{solution} \ \text{to}~(23). \end{array} \right. $$

It is not difficult to check that there exists $C({\varOmega },T,C_{F},\|f\|_{L^{2}(Q)})$ such that, if

$$ \min(\mu_{1},\mu_{2}) > C({\varOmega},T,C_{F},\|f\|_{L^{2}(Q)}), $$

the mapping Λ is a contraction. Indeed, from the usual energy estimates, setting $\mu _{0} := \min \limits (\mu _{1},\mu _{2})$, it is clear that:

$$ \begin{array}{@{}rcl@{}} \| {\varLambda}(v) - {\varLambda}(\tilde v) \|_{\mathcal{U}}^{2}& \leq& \sum\limits_{i=1}^{2} \frac{1}{{\mu_{i}}^{2}} \iint_{\mathcal{O}_{i} \times (0,T)} | (\phi_{1},\phi_{2}) - (\tilde \phi_{1}, \tilde \phi_{2}) |^{2} dx dt \\ \noalign{} \phantom{\| {\varLambda}(v) - {\varLambda}(\tilde v) \|_{\mathcal{U}}^{2}} &\leq& {\mu_{0}}^{-2} C({\varOmega},T) \| (\phi_{1},\phi_{2}) - (\tilde \phi_{1}, \tilde \phi_{2}) \|_{L^{2}(Q) \times L^{2}(Q)}^{2} \\ \noalign{} \phantom{\| {\varLambda}(v) - {\varLambda}(\tilde v) \|_{\mathcal{U}}^{2}} &\leq& {\mu_{0}}^{-2} C({\varOmega},T,C_{F}) \| y - \tilde y \|_{L^{2}(Q)}^{2} \\ \noalign{} \phantom{\| {\varLambda}(v) - {\varLambda}(\tilde v) \|_{\mathcal{U}}^{2}} &\leq& {\mu_{0}}^{-2} C({\varOmega},T,C_{F},\|f\|_{L^{2}(Q)}) \| v - \tilde v \|_{\mathcal{U}}^{2} \end{array} $$

for all $v, \tilde v \in \mathcal {U}$, where the notation is self-explanatory.

As a consequence, we find that, if μ₁ and μ₂ are large enough, (23)–(25) possesses a unique solution. In other words, (6) is uniquely solvable.

Remark 5

For any fixed λ ∈ (0, 1), we can also consider the system (28)–(30). Arguing in a similar way, we can deduce that there exists $C({\varOmega },T,C_{F},\|f\|_{L^{2}(Q)},\lambda )$ such that, for greater values of $\min \limits (\mu _{1},\mu _{2})$, this system possesses exactly one solution.