114 research outputs found
Minimum energy with infinite horizon: From stationary to non-stationary states
No full textWe study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state x̄=0 (at time t=−∞) into an arbitrary non-stationary state x (at time t=0). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state x into the equilibrium state x̄=0). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model
A trace regularity result for thermoelastic equations with application to optimal boundary control
No full textAbstractWe consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations
A Nullstellensatz for Lojasiewicz ideals
No full textFor an ideal of smooth functions a that is either łojasiewicz or weakly łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety I(Z(a)) in terms of the global łojasiewicz radical and Whitney closure. We also prove that the łojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to revisit Nullstellensatz results due to Bochnak and Adkins-Leahy and to resolve positively a modification of the Nullstellensatz conjecture due to Bochnak. © European Mathematical Society
Minimum energy for linear systems with finite horizon: a non-standard Riccati equation
No full textThis paper deals with a non-standard infinite dimensional linear quadratic control problem arising in the physics of non-stationary states (see, for example, Bertini et al. J Statist Phys 116:831â841, 2004): finding the minimum energy to drive a fixed stationary state x ̄ = 0 into an arbitrary non-stationary state x. The Riccati equation (RE) associated with this problem is not standard since the sign of the linear part is opposite to the usual one, thus preventing the use of the known theory. Here we consider the finite horizon case when the leading semigroup is exponentially stable. We prove that the linear selfadjoint operator P(t), associated with the value function, solves the above-mentioned RE (Theorem 4.12). Uniqueness does not hold in general, but we are able to prove a partial uniqueness result in the class of invertible operators (Theorem 4.13). In the special case where the involved operators commute, a more detailed analysis of the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of applications are given
A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control
No full textWe study the infinite horizon linear-quadratic (LQ) problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of partial differential equations (PDEs) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall predominant hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike in the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic components of the dynamics was introduced by the authors in 2005, and a full theory of the LQ-problem on a finite time horizon has been developed. The present paper provides the infinite time horizon theory, culminating in well-posedness of the corresponding (algebraic) Riccati equations. New technical challenges are encountered and new tools are needed, especially in order to pinpoint the differentiability of the optimal solution. The theory is illustrated by means of a boundary control problem arising in thermoelasticity. © 2013 Society for Industrial and Applied Mathematics
Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs
No full textWe study the quadratic optimal control problem over a finite time horizon for a class of abstract systems with non analytic underlying semigroup etA and unbounded control operator B. It is assumed that a suitable decomposition of the operator B*etA* is valid, where only one component satisfies a \u27singular estimate\u27, whereas for the other component specific regularity properties hold. Under these conditions, we prove well posedness of the associated differential Riccati equation, and in particular that the gain operator is bounded on a dense set. In spite of the unifying abstract framework used, the prime motivation (and application) of the resulting theory of linear-quadratic problems comes from optimal boundary control of a thermoelastic system with clamped boundary conditions. The non-trivial trace regularity estimate showing that this PDE mixed problem fits into the distinct class of models under examination-for which we have developed the present, novel optimal control theory-is established, as well
The Positivstellensatz for definable functions on o-minimal structures
No full textIn this note we prove two Positivstellensatze for definable functions of class C-r, 0 less than or equal to r < &INFIN;, in any o-minimal structure S expanding a real closed field R. Namely, we characterize the definable functions that are nonnegative (resp. strictly positive) on basic definable sets of the form F = {f(1) &GE; 0,...,f(k) &GE; 0}.GNSAGADGESDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu
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