A one-dimensional initial-boundary value problem for a thermoelastic beam that is clamped at one end and moves freely between two obstacles at the other, is studied using a method of functional analysis. The boundary conditions at the free end are represented either by classic Signorini's conditions or a normal compliance condition. The main result is that the energy associated with the system decays exponentially as time goes to infinity. Numerical examples are included.

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We study environmental pollution problems by using the optimal control theory applied to partial differential equations. We consider the problem to find the optimal way to eliminate pollution in the time, such that the concentration is close to a standard level which does not affect the ecological equilibrium when the source is pointwise. We consider a quadratic cost functional and we prove the existence and uniqueness of optimal control. We find a characterization which makes possible the computing of optimal control. Additionally, we consider the problem moving the pointwise source. So we define a functional j(b) that associates to any point b in a region of the space the optimal cost functional applied to the optimal control. We show that j(b) is differentiable, provided that the controls are taken in a convenient subset of admissible functions satisfying the cone properties. We also find the point in the region, for which the cost functional is minimum.