Stefan Volkwein

Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional assumptions on the solution of the state equation, the differentiability of the value function with respect to the initial time. In our proof, we rely on local growth assumptions commonly associated with the study of second-order sufficient conditions. These assumptions are generally applicable to a wide range of problems, including, for instance, certain tracking-type problems. Finally, we discuss the differentiability of the value function in a neighborhood of the optimal trajectory when a growth condition for optimal controls is used.

District heating networks (DHNs) offer significant potential for improving energy efficiencythrough optimization-based control. Lowering the source feed temperature reduces heat losses,but the feed temperature at consumers must remain above a prescribed minimum threshold. Thecooling of water in the network and, therefore, the consumer temperatures depend on demand-driven flow conditions. Because future heat demand is uncertain and forecast errors increasewith the prediction horizon, controls optimized using forecast data may violate the minimumtemperature constraint in practice. This study demonstrates the advantages of an model predictivecontrol (MPC) approach that combines demand forecasting, dynamic simulation, and repeatedre-optimization of the source feed temperature control signal in a DHN.Using the dynamic thermo-hydraulic network simulator AD Net Heat, we compare a 24-hour open-loop optimization with an MPC optimization strategy that re-optimizes the controlsignal hourly using updated demand forecasts. We identify an optimal MPC prediction horizonfor the considered network and evaluate both strategies on real demand data to quantifythe advantages of the MPC optimization strategy with respect to violations of the minimumtemperature constraint. For a representative day, the MPC optimization strategy reduces theduration of minimum-temperature violations at consumers by nearly one third and the maximumviolation by about one half compared with the open-loop optimization strategy. An evaluationover 30 consecutive days confirms a substantial reduction in severe violations of the minimumtemperature constraint at consumers. The results show that combining demand forecasting,dynamic simulation, and MPC improves the robustness of district heating operation underdemand uncertainty.

Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this research question within the framework of inverse problems. In this work, we consider optimization techniques to solve the inverse problem using Tikhonov regularization and data from laboratory experiments. We propose an iterative strategy that eliminates the need for further laboratory experiments. Our method is applied to identify the coupling and damping coefficients in a system of oscillators, ensuring an efficient and experiment-free approach. We present our results and compare them with those obtained from an alternative, purely experimental approach. By employing our proposed strategy, we demonstrate a significant reduction in the number of laboratory experiments required.

We present a model predictive control () framework for linear switched evolution equations arising from a parabolic partial differential equation (). First-order optimality conditions for the resulting finite-horizon optimal control problems are derived. This analysis allows for the incorporation of convex control constraints and sparse regularization. Then, to mitigate the computational burden of the procedure, we employ Galerkin reduced-order modeling () techniques to obtain a low-dimensional surrogate for the state-adjoint systems. We derive recursive a posteriori estimates for the feedback law and the - closed-loop state and show that the - trajectory evolves within a neighborhood of the true trajectory, whose size can be explicitly computed and is controlled by the quality of the . Such estimates are then used to formulate two - algorithms with closed-loop certification.

Synthetic gene circuits provide experimentally tractable systems for dissecting how genetic feedback and diffusible signals generate multicellular patterns. However, building multi-component circuits whose behavior can be quantitatively linked to module-level measurements, diffusion, and spatial boundary conditions remains challenging. Here, we designed and engineered a bacterial patterning system in which positive feedback, delayed negative feedback, and two orthogonal quorum-sensing signals are integrated in Escherichia coli . We first implemented and characterized the feedback modules separately, measured the effective diffusion of the signals in the experimental setup, and used these data to parameterize a mathematical model. In quasi-2D bacterial lawns, the complete circuit generated self-organized spatiotemporal dynamics consisting of an sfGFP activation front followed by successive mCherry propagating pulses/traveling waves. Model-guided perturbations showed that lawn size, lawn position relative to the domain boundary, and signal degradation modulate the timing, amplitude, wavelength, and directionality of these patterns. Our work establishes a modular synthetic multicellular reaction-diffusion system in which circuit architecture, signal diffusion, and boundary-mediated signal exchange can be experimentally connected to emergent patterning dynamics.

We analyze a pointwise tracking multiobjective optimal control problem subject to the Poisson problem and bilateral control constraints. To approximate Pareto optimal points and the Pareto front numerically, we consider two different finite element-based scalarization techniques, namely the weighted-sum method and the reference point method, where in both methods many scalar-constrained optimization problems have to be solved. We prove a priori error estimates for both scalarizations. The underlying subproblems of either method are solved with a Barzilai-Borwein gradient method. Numerical experiments illustrate the accuracy of the proposed method.

We address the stabilization of linear, time-varying parabolic PDEs using finite-dimensional receding horizon controls (RHCs) derived from reduced-order models (ROMs). We first prove exponential stability and suboptimality of the continuous-time full-order model (FOM) RHC scheme in Hilbert spaces. A Galerkin model reduction is then introduced, along with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This results in a ROM-based RHC algorithm that adaptively constructs reduced-order controls, ensuring exponential stability of the FOM closed-loop state and providing computable performance bounds with respect to the infinite-horizon FOM control problem. Numerical experiments with a non-smooth cost functional involving the squared 𝓁 ¹ -norm confirm the method's effectiveness, even for exponentially unstable systems.

Accurate parameter dependent electro-chemical numerical models for lithium-ion batteries are essential in industrial application. However, some parameters of each battery cell are unknown, so that a parameter estimation is necessary to infer them. The field of optimal input/experimental design deals with creating an optimal experimental settings facilitating the estimation problem. Here we apply two different input design algorithms that aim at maximizing the observability of the true, unknown parameters. As the design algorithms are built independent of the model, the same results and motivation are applicable to more complex battery cell models and, moreover, to other applications.

In this work, we present an algorithm for computing an enclosure of the nondominated set of multi-objective mixed-integer nonconvex optimization problems. In contrast to existing solvers for this type of problem, this algorithm is not based on a branch-and-bound scheme but rather relies on a relax-and-refine approach. While this is an established technique in single-objective optimization, several adaptions to the multi-objective setting have to be made in order to exploit the full potential of this idea. To that end, we propose an intensified individualization of the relaxations to the respective parts in the image space resulting in a novel adaptive box-based relaxation technique for nonconvex terms. We provide numerical tests for the new algorithm that show both its strengths and limitations.

We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss–Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution (Kartmann et al 2024 Comput. Sci. Eng. 1 3) for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use.