The main aim of this paper is to develop a theory for non-autonomous parabolic equations with time-dependent measures on the spatial domain appearing as right hand sides. Restricting these measures to ones which have their supports on 'curves' or 'surfaces' - the latter understood in the sense of geometric measure theory - we succeed in interpreting them as distributional objects from a (negative indexed) Sobolev-Slobodetskii space Ws,2(Omega) with s close to -1. For these indices s a tailor suited parabolic theory is established, based on results of Disser et al. [Ann. Sci. Norm. Super. Pisa, Cl. Sci. 17 (2017) 65-79] and Haller-Dintelmann et al. [Ann. Mat. Pura Appl. 198 (2019) 1227-1241]. The proposed frame work is well-suited for optimal control problems with controls acting on sub-manifolds.

This work presents a digital twin framework for output-feedback stabilization and parameter identification in uncertain dynamical systems. A virtual model evolves in parallel with the physical process, assimilating measurement data in real time. By design, the digital twin reconstructs the system state and generates a stabilizing feedback, while model parameters are simultaneously inferred from data of the controlled dynamics using a Bayesian approach. Numerical results for the coupled physical-virtual dynamics demonstrate how digital twins can act jointly as observers, parameter estimators, and control agents, ensuring robust performance under uncertainty.

This work concentrates on a class of Neumann optimal control problems for semilinear parabolic equations on an infinite horizon domain $Q = \Omega \times (0,\infty )$ subject to a control constraint of the form $\alpha \leq u(x,t) \leq \beta $ for $(x,t) \in \Gamma \times (0,\infty )$, where Γ is the boundary of Ω. Existence of a solution, first- and second-order optimality conditions are established. Finally, the approximation of the solution by finite horizon control problems is addressed and some error estimates are provided.

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

The Kalman(-Bucy) filter is the natural choice for the state reconstruction of disturbed, linear dynamical systems based on flawed and incomplete measurements. Taking a deterministic viewpoint, this work investigates possible extensions of the concept to systems with uncertain dynamics and noise covariances. In a theoretical analysis, error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two example systems, allowing for a comparison of the estimators.

This paper is dedicated to the analysis of infinite horizon optimal control problems subject to semilinear parabolic equations with constraints on the controls and discounted cost functionals. The discount factors on the cost and the state components are allowed to differ from each other. First-order as well as second-order optimality conditions are derived and the importance of allowing different discount factors for the second-order analysis for the class of nonlinearities under consideration is demonstrated. Finally convergence and rate of convergence for the approximation of the infinite horizon problem by a family of finite horizon problems is proven.

It is shown that a switching control involving a finite number of Dirac delta actuators is able to steer the state of a general class of nonautonomous parabolic equations to zero as time increases to infinity. The strategy is based on a recent feedback stabilizability result, which utilizes control forces given by linear combinations of appropriately located Dirac delta distribution actuators. Then, the existence of a stabilizing switching control with no more than one actuator is active at each time instant is established. For the implementation in practice, the stabilization problem is formulated as an infinite-horizon optimal control problem, with cardinality-type control constraints enforcing the switching property. Subsequently, this problem is tackled using a receding horizon framework. Its suboptimality and stabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties.

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including optimal feedback control, game theory, and optimal transport. We leverage the fact that any semiconcave function can be represented as the infimum of a countable family of C^2 functions. This infimum is expressed in a form that allows approximation by finitely many functions, combined with smoothing operations, such that each element of the approximating sequence remains semiconcave. The active sets of indices contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the gradients of the elements in the expansion of the approximating functions form a probability distribution, a property of particular interest for the value function in optimal control. Approximation results are established in C(Ω̅) and in W^1,p(Ω) for p ∈ [1,∞) and p = ∞. Finally, numerical results are presented to illustrate the approach on a test example.

The value function for an infinite horizon tracking type optimal control problem with semilinear parabolic equation is investigated. In view of a possible nonconvexity of the optimal control problem, a local version of the value function is considered. Its differentiability is proved for initial data in a neighborhood around the nominal initial value, provided a second order sufficient optimality condition is fulfilled for the nominal locally optimal control. Based on the differentiability of the value function, a Hamilton-Jacobi-Bellman equation is derived.

A quantitative frequency-domain condition related to the exponential stabilizability for infinite-dimensional linear control systems is presented. It is proven that this condition is necessary and sufficient for the stabilizability of special systems, while it is a necessary condition for the stabilizability in general. Applications are provided.

Stabilization of linear control systems with parameter-dependent system matrices is investigated. A Riccati based feedback mechanism is proposed and analyzed. It is constructed by means of an ensemble of parameters from a training set. This single feedback stabilizes all systems of the training set and also systems in its vicinity. Moreover its suboptimality with respect to optimal feedback for each single parameter from the training set can be quantified.

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests.

The Kalman(-Bucy) filter is the natural choice for the state reconstruction of disturbed, linear dynamical systems based on flawed and incomplete measurements. Taking a deterministic viewpoint this work investigates possible extensions of the concept to systems with uncertain dynamics and noise covariances. In a theoretical analysis error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two example systems allowing for a comparison of the estimators.

Two related numerical schemes for the realization of the Mortensen observer or minimum energy estimator for the state reconstruction of non-linear dynamical systems subject to deterministic disturbances are proposed and compared. Both approaches rely on a polynomial approximation of the value function associated with the energy of the disturbances of the system. Such an approximation is obtained via interpolation considering not only the values but also first and second order derivatives of the value function in a set of sampling points. The scheme is applied to four examples and the results are compared with the well known extended Kalman filter.

The challenge of constructing feedback control laws for risk-averse optimal control of partial differential equations (PDEs) with random coefficients is addressed. The control objective composes a tracking-type cost with the nonlinear entropic risk measure. A sequential quadratic programming scheme is derived that iteratively solves linear quadratic subproblems obtained through second-order Taylor expansions of the objective functional, with each subproblem re-centered at the previous iterate. It is shown that this method converges locally quadratically to the unique risk-averse optimal control. This work provides the first rigorous feedback synthesis for risk-averse objectives subject to PDEs with random coefficients.

A shape optimization problem subject to an elliptic equation in the presence of missing data on the Dirichlet boundary condition is considered. It is formulated by optimizing the deformation field that varies the spatial domain where the Poisson equation is posed. To take into consideration the missing boundary data the problem is formulated as a no-regret problem and approximated by low-regret problems. This approach allows to obtain deformation fields that are robust against the missing information. The formulation of the regret problems was achieved by employing the Fenchel transform. Convergence of the solutions of the low-regret to the no-regret problems is analyzed, the gradient of the cost is characterized and a first order numerical method is proposed. Numerical examples illustrate the robustness of the low-regret deformation fields with respect to missing data. To the best of our knowledge, this is the first time that a numerical investigation is reported on the level of effectiveness of the low-regret approach in the presence of missing data in an optimal control problem.

This paper is devoted to proving L-infinity-estimates for the solution of semilinear parabolic equations. The uniform estimates are obtained on the infinite time interval under the assumption that the solution is square integrable. This setting is useful for stabilization problems formulated as optimal control problems. The inhomogenous forcing function are chosen as elements of anisotropic Lebesgue spaces. Different boundary conditions on bounded domains with a Lipschitz continuous boundary are investigated.

This work addresses the synthesis of optimal feedback control laws via machine learning. In particular, the Averaged Feedback Learning Scheme (AFLS) and a data driven method are considered. Hypotheses for each method ensuring the convergence of the evaluation of the objective function of the underlying control problem at the obtained feedback-laws towards the optimal value function are provided. These hypotheses are connected to the regularity of the value function and the stability of the dynamics. In the case of AFLS these hypotheses only require Hölder continuity of the value function, whereas for the data driven method the value function must be at least C^2. It is demonstrated that these methods are connected via their optimality conditions. Additionally, numerical experiments are provided by applying both methods to a family control problems, parameterized by a positive real number which controls the regularity of the value function. For small parameters the value function is smooth and in contrast for large parameters it is non-differentiable, but semi-concave. The results of the experiments indicate that both methods have a similar performance for the case that the value function is smooth. On the other hand, if the value function is not differentiable, AFLS has a better performance which is consistent with the obtained convergence results.

The stabilization of nonautonomous parabolic equations is achieved by feedback inputs tuning a finite number of actuators, where it is assumed that the input is subject to a time delay. To overcome destabilizing effects of the time delay, the input is based on a prediction of the state at a future time. This prediction is computed depending on a state-estimate at the current time, which in turn is provided by a Luenberger observer. The observer is designed using the output of measurements performed by a finite number of sensors. The asymptotic behavior of the resulting coupled system is investigated. Numerical simulations are presented validating the theoretical findings, including tests showing the response against sensor measurement errors.

Taking a deterministic viewpoint this work investigates extensions of the Kalman-Bucy filter for state reconstruction to systems containing parametric uncertainty in the state operator. The emphasis lies on risk averse designs reducing the probability of large reconstruction errors. In a theoretical analysis error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two examples allowing for a comparison of risk neutral and risk averse estimators.