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A maximum principle for optimally controlled systems of conservation laws. (English) Zbl 0935.49012
The authors consider an optimization problem for strictly hyperbolic systems of the type \[ u_t+ [F(u)]_x= h(t,x,u,z),\quad u(0,x)= \overline u(x), \] with distributed control \(z= z(t,x)\in Z\subset \mathbb{R}^p\), where \(u(t,x)\in \mathbb{R}^m\) while \((t,x)\in [0,T]\times \mathbb{R}\). The cost function has the form \(J(u)= \int_{-\infty}^{\infty} V(x,u(T,x)) dx\).
The main result is the following. Suppose \(h\), \(V\) and \(F\) are \({\mathcal C}^1\) (continuously differentiable) functions and each characteristic field for the system is either linearly degenerate or genuinely nonlinear; let the optimal control \(\widehat z=\widehat z(t,x)\) be a \({\mathcal C}^1\) function, and let the corresponding optimal solution \(\widehat u= \widehat u(t,x)\) be piecewise \({\mathcal C}^1\) and satisfy some additional regularity assumption; define the adjoint vector \((v^*,\xi^*)\) with terminal conditions \(v^*(T, x)= \nabla_uV(x,\widehat u(T,x))\), \(\xi^*_\alpha(T)= \Delta V(x_\alpha(T))\), \(\alpha= 1,\dots, N\).
Then the maximality condition \[ v^*(t,x)\cdot h(t,x,\widehat u(t,x),\widehat z(t,x))= \max_{z\in Z} v^*(t,x)\cdot h(t, x,\widehat u(t,x),z) \] holds at each point \((t,x)\) where both \(v^*\) and \(\widehat u\) are continuous.
The paper does not contain a non-trivial example of a system and a cost function that satisfy all the conditions of the theorem above.

MSC:
49K20 Optimality conditions for problems involving partial differential equations
35L65 Hyperbolic conservation laws
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References:
[1] A. Bressan , Lecture notes on systems of conservation laws , S.I.S.S.A. , Trieste ( 1993 ). · Zbl 0802.35095
[2] A. Bressan - A. Marson , A variational calculus for shock solutions to systems of conservation laws , to appear in Comm. P.D.E. [3] R. Di Perna , Entropy and the uniqueness of solutions to hyperbolic conservation laws , in Nonlinear Evolution Equations , M. Crandall Ed., Academic Press , New York ( 1978 ), pp. 1 - 16 . MR 513809 | Zbl 0469.35064 · Zbl 0469.35064
[3] R. Di Perna , Uniqueness of solutions to hyperbolic conservation laws , Indiana Univ. Math. J. , 28 ( 1979 ), pp. 244 - 257 . MR 523630 | Zbl 0409.35057 · Zbl 0409.35057
[4] Li Ta-Tsien - Yu Wen-Ci , Boundary value problems for quasilinear hyperbolic systems , Duke University Mathemathics Series V ( 1985 ). MR 823237 | Zbl 0627.35001 · Zbl 0627.35001
[5] B. Rozdestvenskii - N. Yanenko , Systems of Quasi-Linear Equations , A. M. S. Translations of Mathematical Monographs , Vol. 55 ( 1983 ). MR 694243 | Zbl 0513.35002 · Zbl 0513.35002
[6] M. Schatzman , Introduction a l’analyse des systemes hyperboliques de lois de conservation non-lineaires , Equipe d’Analyse Numerique Lyon Saint-Etienne , 37 ( 1985 ).
[7] J. Smoller , Shock Waves and Reaction-Diffusion Equations , Springer-Verlag , New York ( 1983 ). MR 688146 | Zbl 0807.35002 · Zbl 0807.35002
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