On some optimal control problem for two-dimensional continuous systems (Q1332460)

The authors consider optimal control problems governed by two-dimensional hyperbolic distributed systems of the form \[ \begin{cases} z_{xy}= A_ 0 z+ A_ 1 z_ x+ A_ 2 z_ y+ B_ 0 u_ 0+ B_ 1 u_ 1+ B_ 2 u_ 2,\\ z(x,0)= z(0,y)= 0,\end{cases} \tag{1} \] where \(A_ i\) and \(B_ i\) \((i= 0,1,2)\) are matrices and \((x,y)\in Q= [0,1]\times [0,1]\). The cost functional is \[ I(z,u)= \int_ Q (ax+ b_ 0 u_ 0+ b_ 1 u_ 1+ b_ 2 u_ 2)dx dy+ \sum^ 2_{i=0} \beta_ i S_ i(u_ i), \] where \(a\) and \(b_ i\) \((i=0,1,2)\) are vectors, \(\beta_ i\) are positive numbers, and \(S_ i\) are functionals which take into account the rate of change of the controls \(u_ 0\), \(u_ 1\), \(u_ 2\). The main result consists in the determination of the necessary conditions of optimality, provided equation (1) has a unique solution \(z\) for each control \((u_ 0,u_ 1,u_ 2)\). Moreover, particular cases as the one with constant coefficients are studied in detail.