Eigenvalue bounds for the Gramian operator of the heat equation - Lazar, Martin; Zuazua Iriondo, Enrique (Automatica, 2024)
This paper is concerned with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly focus on the Gramian operator, which arises in the context of control and observation of heat processes in infinite time, which is normally the first step towards observations in a finite time horizon. By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results

Greedy search of optimal approximate solutions - Lazar, Martin; Zuazua Iriondo, Enrique (Pure and Applied Functional Analysis, 2023)
In this paper we develop a procedure to deal with a family of parameter-dependent ill-posed problems, for which the exact solution in general does not exist. The original problems are relaxed by considering corresponding approximate ones, whose optimal solutions are well defined, where the optimality is determined by the minimal norm requirement. The procedure is based upon greedy algorithms that preserve, at least asymptotically, Kolmogorov approximation rates. In order to provide a-priori estimates for the algorithm, a Tychonoff-type regularization is applied, which adds an additional parameter to the model. The theory is developed in an abstract theoretical framework that allows its application to different kinds of problems. We present a specific example that considers a family of ill-posed elliptic problems. The required general assumptions in this case translate to rather natural uniform lower and upper bounds on coefficients of the considered operators.<br
Keywords: Optimal approximate solutions; parametrized PDEs; greedy algorithms

Turnpike properties for partially uncontrollable systems - Gugat, Martin; Lazar, Martin (Automatica, 2023)
We analyse the turnpike properties for a general, infinite dimensional, linear–quadratic (LQ) optimal control problem, in both the deterministic and the parameter dependent case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for optimal averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result that extends the averaged controllability approach to optimal control problems In both cases, the results do not require assumptions on stabilizability and detectability, which are most commonly used in the study of turnpike phenomena.
Keywords: Turnpike phenomenon; Optimal averaged control; Infinite-time admissibility; Infinite-dimensional systems; Optimization under uncertainties