Variational maximum principle in controlled systems of one-dimensional hyperbolic equations (Q1589144)

The paper considers optimal control problems for the semilinear hyperbolic system \[ z_t+ A(x, t)z_x= f(x,t,z,u)\tag{1} \] in a rectangle \(D= \{(x, t)\in\mathbb{R}^2\mid 0< x< a, 0< t< T\}\). The cost functional is standard and includes an additional integral along a line \(\{(x, t)\mid x= x_0, 0< t< T\}\) with \(x_0\subset (0,a)\). The author gives short proofs for the Pontryagin's maximum principle and its analogue along characteristics of (1).