This paper presents two observability inequalities for the heat equation over $\Omega \times (0,T)$. In the first one, the observation is from a subset of positive measure in $\Omega \times (0,T)$, while in the second, the observation is from a subset of positive surface measure on $\partial \Omega \times (0,T)$. It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Bellman systems corresponding to stochastic differential games arising from a cost functional which models risk aspects are considered. Here it leads to diagonal elliptic systems without zero order term so that no simple $L^{\infty }$-estimate is available.

One establishes some convexity criteria for sets in $R^2$. They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.

This Note is the Part II of a previous Note with the same title. One refers to holonomic systems $\mathrm{\Sigma }=\mathcal{A}\bigcup \mathcal{U}$ with two degrees of freedom, where the part $\mathcal{A}$ can schemetize a swing or a pair of skis and $\mathcal{U}$ schemetizes whom uses $\mathcal{A}$. The behaviour of $\mathcal{U}$ is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on $\mathrm{\Sigma }$ minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition $\mathrm{\Gamma }$ which implies...

See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\mathrm{\Sigma }$, $\chi$ and $M$ satisfy conditions (11.7) when $𝒬$ is a polynomial in $\dot{\gamma }$, conditions (C)-i.e. (11.8) and (11.7) with $𝒬\equiv 0$-are proved to be necessary for treating satisfactorily $\mathrm{\Sigma }$'s hyper-impulsive motions (in which positions can suffer first order discontinuities)....

In [1] I and II various equivalence theorems are proved; e.g. an ODE $(ℰ)\dot{z}=F(t,z,u,\dot{u})(\in {ℝ}^m)$ with a scalar control $u=u(\cdot )$ is linear w.r.t. $\dot{u}$ iff $(\alpha )$ its solution $z(u,\cdot )$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta )$

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain with control distributed in an arbitrary fixed subdomain. The result that we obtain in this paper is as follows. Suppose that we have a given stationary point of the Navier-Stokes equations and our initial condition is sufficiently close to it. Then there exists a locally distributed control such that in a given moment of time the solution of the Navier-Stokes...

In this paper we discuss the exact null controllability of linear as well as nonlinear Black–Scholes equation when both the stock volatility and risk-free interest rate influence the stock price but they are not known with certainty while the control is distributed over a subdomain. The proof of the linear problem relies on a Carleman estimate and observability inequality for its own dual problem and that of the nonlinear one relies on the infinite dimensional Kakutani fixed point theorem with $L^2$...

A hybrid flexible beam equation with harmonic disturbance at the end where a rigid tip body is attached is considered. A simple motor torque feedback control is designed for which only the measured time-dependent angle of rotation and its velocity are utilized. It is shown that this control can impel the amplitude of the attached rigid tip body tending to zero as time goes to infinity.

A hybrid flexible beam equation with harmonic disturbance at the end where a rigid tip body is attached is considered. A simple motor torque feedback control is designed for which only the measured time-dependent angle of rotation and its velocity are utilized. It is shown that this control can impel the amplitude of the attached rigid tip body tending to zero as time goes to infinity.

In the paper, the problem of source function reconstruction in a differential equation of the parabolic type is investigated. Using the semigroup representation of trajectories of dynamical systems, we build a finite-step iterative procedure for solving this problem. The algorithm originates from the theory of closed-loop control (the method of extremal shift). At every step of the algorithm, the sum of a quality criterion and a linear penalty term is minimized. This procedure is robust to perturbations...

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot )$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot )$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\mathrm{\Sigma }$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy conditions,...

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega \subset {ℝ}^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega$ of Lebesgue measure $\left|\omega \right|=L\left|\Omega \right|$, where $L\in (0,1)$ is fixed. We denote by ${𝒰}_L$ the class of all possible such subsets. Let $T\>0$. We consider first the benchmark problem of maximizing...