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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:
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