## Approximation of an inverse problem for variational inequalities.(English)Zbl 0841.65054

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.
Reviewer: L.Bakule (Praha)

### MSC:

 65K10 Numerical optimization and variational techniques 49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000) 49J40 Variational inequalities 49N45 Inverse problems in optimal control