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.
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Acknowledgments
This paper was partially written during a stay of the first author at the Institute of Mathematics of the University of Sevilla (IMUS). He is indebted to this Institute for its assistance.
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Communicated by: Enrique Zuazua
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Appendix: Existence and uniqueness of a solution to (6)
Appendix: Existence and uniqueness of a solution to (6)
Let us consider the mapping:
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:
Let us identify the mapping \({\mathscr{A}}\). For every \((v_{1}, v_{2}) \in \mathcal {U}\), one has:
where the ϕi are given by
and y is the solution to
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:
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:
If y and z denote the solutions to (36) corresponding to (v1,v2) and (w1,w2), we have:
Thus,
Therefore, \(\mathcal {A}\) is self-adjoint.
For completeness, let us recall the proof of strong positiveness. Thus, note that, for any \((v_{1},v_{2}) \in \mathcal {U}\):
Let z1 (resp. z2) be the solution to (36) with v2 = 0 (resp. v1 = 0). Then, from standard integration by parts, we see that:
Consequently, there exists C0 depending on Ω and T such that:
From (41) and (43), we see that, if \(\min \limits (\mu _{1},\mu _{2}) > \sqrt {2} C_{0}\), then
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
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 μi.
Let us identify b, that is, the nonhomogeneous term in the affine mapping (34). One has:
where \(\bar {\phi }_{i}\) is the solution to
and \(\bar {y}\) is the solution of
Now, if we define by
and by
we deduce that (6) is equivalent to
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 μi 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:
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 CF > 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
It is not difficult to check that there exists \(C({\varOmega },T,C_{F},\|f\|_{L^{2}(Q)})\) such that, if
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:
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.
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de Carvalho, P., Fernández-Cara, E. & Ferrel, J.B.L. On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation. Adv Comput Math 46, 73 (2020). https://doi.org/10.1007/s10444-020-09812-z
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DOI: https://doi.org/10.1007/s10444-020-09812-z
Keywords
- Wave equation
- Finite elements and finite differences
- Bi-objective optimal control
- Nash and Pareto equilibria