Abstract. Problems of internal and pointwise observation and control for the one-dimensional wave equation arise in the simulation of control and identification processes in electrical engineering, flaw detection, and medical tomography. The generally accepted way of modelling sensors and actuators as pointlike objects leads to results which may make no apparent physical sense: they may depend, for instance, on the rationality or irrationality of the location for a point sensor or actuator. We propose a new formulation of sensor (actuator) action, expressed mathematically by using somewhat unconventional spaces for data presentation and processing. For interaction restricted to an interval of length e , the limit system of observation (or control) now makes sense when e tends to zero without a sensitive dependence on the precise location of the limiting point.

Abstract. We consider the exact null controllability problem for the semi- linear heat equation with dissipative nonlinearity in a bounded domain of R n . The main result of the article asserts that if the nonlinearity is even mildly superlinear, then global null controllability in an arbitrarily short time fails; instead we provide sharp estimates for the controllability time in terms of the size of the initial data.

One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution of the approximating random walk.

Geometric versions of a class of Gaussian processes are investigated as possible models for stock prices. Arbitrage opportunities are constructed for these processes showing that option pricing is not possible with these models.

We study the Lagrange Problem of Optimal Control with a functional \(\int_{a}^{b}L\left(t,\,x\left(t\right),\,u\left(t\right)\right)\,dt\) and control-affine dynamics \(\dot{x}\)= f(t,x) + g(t,x)u and (a priori) unconstrained control u∈ \bf R m . We obtain conditions under which the minimizing controls of the problem are bounded—a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.

We prove the existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (X;μ ) , where X is a (possibly infinite-dimensional) space. Our approach is purely analytic based on the theory of sectorial forms. The generators covered are, e.g., small perturbations (in the sense of sectorial forms) of operators generating hypercontractive semigroups. An essential ingredient of the proofs is a new result on compact embeddings of weighted Sobolev spaces H 1,2 (ρ . . . dx) on \bf R d (resp. a Riemannian manifold) into L 2 (ρ dx) . Probabilistic consequences are also briefly discussed.

Mascarenhas gave an instance of linear programming problems to show that the long-step affine scaling algorithm can fail to converge to an optimal solution with the step-size λ=0.999 . In this note, we give a simple and clear geometrical explanation for this phenomenon in terms of the Newton barrier flow induced by projecting the homogeneous affine scaling vector field conically onto a hyperplane where the objective function is constant. Based on this interpretation, we show that the algorithm can fail for "any" λ greater than about 0.91 (a more precise value is 0.91071), which is considerably shorter than λ = 0.95 and 0.99 recommended for efficient implementations.

We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of an epicontingent solution, or a viscosity supersolution to a partial differential equation called Isaacs' equation. Links between supersolutions and epicontingent solutions to Isaacs' equation are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H ∞ problem from contingent formulation of Isaacs' equation.

In this paper we derive optimality conditions in the form of a maximum principle for an optimal control problem for nonlinear implicit, or descriptor, control systems. The regularity conditions which are imposed on the system allow us to reduce the optimal control problem to an equivalent nonsmooth variational one. Nonsmooth analysis techniques together with the sliding variations of relaxed controls from optimal control are used to obtain necessary conditions.