Approximation of an inverse problem for variational inequalities (Q1901108)

Consider a hyperbolic system described by the variational inequalities \[ (\ddot x(t) - Bu(t) - f(t), \dot x(t) - z)_H + \langle Ax(t), \dot x(t) - z\rangle + \varphi(\dot x(t)) - \varphi(z) \leq 0,\tag{1} \] \(t \in T = [t_0,\theta]\), \(\forall z \in V\), \(x(t_0) = x_{10} \in V\), \(\dot x(t_0) = x_0 \in H\), where \(H\), \(V\) are real Hilbert spaces with norms \(|.|_H\), \(|.|_V\), resp. \(H = H^*\), \(V \subset H\), \(V\) is densely and continuously imbedded in \(H\), \((.,.)_H\) is the scalar product in \(H\), \(\langle .,.\rangle\) is the duality between \(V\) and \(V^*\), \(f(.) \in L_2(T;H)\) is a given function, \(A : V \to V^*\) is a linear continuous \((A \in L(V;V^*))\) symmetric operator satisfying \(\langle Ay,y \rangle \geq \omega |y|^2_V\), \(\forall y \in V\) for certain \(\omega > 0\). \((U,|.|_U)\) is a uniformly convex Banach space, \(\varphi : H \to \mathbb{R}^+ \cup \{+\infty\}\) is a convex lower semicontinuous proper function, \(0 \in D(\varphi) = \{x \in H : \varphi(x) < + \infty\}\), \(B \in L(U;V)\) is a linear continuous operator, \(u(t) \in P\) for almost every \(t \in T\), \(P \subset U\) is a convex closed bounded set. Suppose that for any control \(u(.) \in P(.) = \{u(.) \in L_2(T;U) : u(t) \in P\), \(t \in T\}\) there exists a single solution of the system (1), \(x(.) = x(.;t_0,x_0,x_{10},u(.)) \in C(T,V)\), such that \(\dot x(.) \in C(T;H) \cap L_\infty(T;V)\), \(\ddot x(.) \in L_2(T;H)\). The motion \(x(.)\) depending on an unknown control \(u(.) \in P\) proceeds at the time interval \(T\), is unknown. Only \(\dot {x}(\tau_i)\) are approximately measured at the time instants \(\tau_i \in \Delta = \{\tau_i\}^m_{i = 0}\), \(\tau_{i + 1} = \tau_i + \delta\), \(\tau_0 = t_0\), \(\tau_m = \theta\). Let \(U_*(x(.))\) be the set of all control from \(P(.)\) generating \(x(.)\). The problem is to calculate an approximation to a certain element \(u(.) \in U_*(x(.))\) synchronously with the process using nonaccurate measurements of \(\dot x(\tau_i)\). An algorithm solving the problem, which is stable against informational and computational perturbations is presented. The solution is based on the theory of position control and utilizes finite-dimensional models.