Searcharxiv⌕ Search

arXiv subjects

Karl Kunisch

Publications and source records attributed to Karl Kunisch.

At least 19 recordsLinked to original sources
arXiv2026

Dynamic output-feedback stabilization of uncertain linear dynamics via digital twins

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.

math.OC↗
arXiv2026

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

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.

math.OC↗
arXiv2026

Infinite Horizon Optimal Control Problems with Discount Factors

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.

math.OC↗
arXiv2026

Structure Preserving Approximation of Semiconcave Functions

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 \(C(\bar \Omega)\) and in \(W^{1,p}(\Omega)\) for \(p \in [1,\infty)\) and \(p = \infty\). Finally, {numerical results} are presented to illustrate the approach on a test example.

math.OC↗
arXiv2025

Stabilization of nonautonomous linear parabolic equations with inputs subject to time-delay

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.

math.OC↗
arXiv2025

Risk averse deterministic Kalman filters for uncertain dynamical systems

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.

math.OC↗
arXiv2025

Approximation of risk-averse optimal feedback control

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.

math.OC↗
arXiv2025

Deterministic Kalman filters for uncertain dynamical systems

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.

math.DS↗
arXiv2025

On feedback stabilisation for the Cahn-Hilliard equation and its numerical approximation

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.

math.OC↗
arXiv2025

Optimal control of a nonlinear kinetic Fokker-Planck equation

A tracking type optimal control problem for a nonlinear and nonlocal kinetic Fokker-Planck equation which arises as the mean field limit of an interacting particle systems that is subject to distance dependent random fluctuations is studied. As the equation of interest is only hypocoercive and the control operator is unbounded with respect to the canonical state space, classical variational solution techniques cannot be utilized directly. Instead, the concept of admissible control operators is employed. For the underlying nonlinearities, local Lipschitz estimates are derived and subsequently used within a fixed point argument to obtain local existence of solutions. Again, due to hypocoercivity, existence of optimal controls requires non standard techniques as (compensated) compactness arguments are not readily available.

math.OC↗
arXiv2024

Low-regret shape optimization in the presence of missing Dirichlet data

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 which 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 analysed, 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. This is likely the first time that a numerical investigation is reported on for the level of effectiveness of the low-regret approach in the presence of missing data in an optimal control problem.

math.OC↗
arXiv2024

Quasi-Monte Carlo integration for feedback control under uncertainty

A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen--Lo\`eve expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC methods.

math.OC↗
arXiv2024

Convergence of machine learning methods for feedback control laws: averaged feedback learning scheme and data driven methods

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 H\"older 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.

math.OC↗
arXiv2024

Stabilizability of parabolic equations by switching controls based on point actuators

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 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 stabilizabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties.

math.OC↗
arXiv2024

Second order analysis for the optimal selection of time delays

For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated optimization problem for the time delay. A first- and second-order sensitivity analysis is performed including an adjoint calculus that avoids the second derivative of the state with respect to the delay.

math.OC↗
arXiv2024

Optimal actuator design based on shape calculus

An approach to optimal actuator design based on shape and topology optimisation techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology.

math.OC↗
arXiv2024

Smooth approximation of feedback laws for infinite horizon control problems with non-smooth value functions

In this work the synthesis of approximate optimal and smooth feedback laws for infinite horizon optimal control problems is addressed. In this regards, $L^{p}$ type error bounds of the approximating smooth feedback laws are derived, depending on either the $C^1$ norm of the value function or its semi-concavity. These error bounds combined with the existence of a Lyapunov type function are used to prove the existence of an approximate optimal sequence of smooth feedback laws. Moreover, we extend this result to the Hölder continuous case by a diagonalization argument combined with the Moreau envelope. It is foreseen that these error bounds could be applied to study the convergence of synthesis of feedback laws via data driven machine learning methods. Additionally, we provide an example of an infinite horizon optimal control problem for which the value functions is non-differentiable but Lipschitz continuous. We point out that in this example no restrictions on either the controls or the trajectories are assumed.

math.OC↗