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The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem subject to the H 1 -critical defocusing semilinear wave equation on a smooth and bounded domain in three spatial dimensions. Due to the criticality of the nonlinearity in the wave equation, unique solutions to the PDE obeying energy bounds are only obtained in special function spaces related to Strichartz estimates and the nonlinearity. The optimal control problem is complemented by pointwise-in-time constraints of Trust-Region type ‖ u (t) ‖ L 2 (Ω) ≤ ω (t). We prove existence of globally optimal solutions to the optimal control problem and give optimality conditions of both first- and second order necessary as well as second order sufficient type. A nonsmooth regularization term for the natural control space L 1 (0 , T ; L 2 (Ω)) , which also promotes sparsity in time of an optimal control, is used in the objective functional. [ABSTRACT FROM AUTHOR]

We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reducedorder model, we construct a robust feedback control based on the \scrH \infty control method, which requires the solution of an associated high-dimensional Hamilton--Jacobi--Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension d \approx 12. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modeling framework for the robust control of PDEs under parametric uncertainties. [ABSTRACT FROM AUTHOR]

An inverse breaking line identification problem formulated as an optimal control problem with a suitable PDE constraint is studied. The constraint is a boundary value problem describing the anti-plane equilibrium of an elastic body with a stress-free breaking line under the action of a traction force at the boundary. The behavior of the displacement is observed on a subset of the boundary, and the optimal breaking line is identified by minimizing the L 2 {L^{2}} -distance between the displacement and the observation. Then the optimal control problem is solved by shape optimization techniques via a Lagrangian approach. Several numerical experiments are carried out to show its performance in diverse situations. [ABSTRACT FROM AUTHOR]

The Receding Horizon Control (RHC) strategy consists in replacing an infinite-horizon stabilization problem by a sequence of finite-horizon optimal control problems, which are numerically more tractable. The dynamic programming principle ensures that if the finite-horizon problems are formulated with the exact value function as a terminal penalty function, then the RHC method generates an optimal control. This article deals with the case where the terminal cost function is chosen as a cut-off Taylor approximation of the value function. The main result is an error rate estimate for the control generated by such a method, when compared with the optimal control. The obtained estimate is of the same order as the employed Taylor approximation and decreases at an exponential rate with respect to the prediction horizon. To illustrate the methodology, the article focuses on a class of bilinear optimal control problems in infinite-dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

A class of infinite horizon optimal control problems involving mixed quasi-norms of Lp-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization. [ABSTRACT FROM AUTHOR]

A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters. [ABSTRACT FROM AUTHOR]

Nonsmooth nonconvex optimization problems involving the ℓp quasi-norm, p∈(0,1], of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction. [ABSTRACT FROM AUTHOR]

A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and a time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided. [ABSTRACT FROM AUTHOR]

An infinite-horizon optimal control problem subject to an infinite-dimensional state equation with state and control variables appearing in a bilinear form is investigated. A sensitivity analysis with respect to the initial condition is carried out. We show in particular that the value function is infinitely differentiable in the neighborhood of the steady state, under a stabilizability assumption. In a second part, we derive error estimates for controls generated by polynomial feedback laws, which are derived from Taylor expansions of the value function. [ABSTRACT FROM AUTHOR]

First and second order sensitivities of the value function in optimal control problems constrained by semilinear parabolic equations with respect to the initial condition are considered. Sufficient conditions are given which allow to express these sensitivities in terms of the adjoint equation and a suitably defined Riccati equation. [ABSTRACT FROM AUTHOR]

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium. [ABSTRACT FROM AUTHOR]

Abstract: An optimal control problem with a time‐parameter is considered. The functional to be optimized includes the maximum over time‐horizon reached by a function of the state variable, and so an L ∞‐term. In addition to the classical control function, the time at which this maximum is reached is considered as a free parameter. The problem couples the behavior of the state and the control, with this time‐parameter. A change of variable is introduced to derive first and second‐order optimality conditions. This allows the implementation of a Newton method. Numerical simulations are developed, for selected ordinary differential equations and a partial differential equation, which illustrate the influence of the additional parameter and the original motivation. [ABSTRACT FROM AUTHOR]

A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of L1 or measure form with respect to frequency but smooth with respect to time. We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only finitely many frequencies. These results cover important systems of physical interest, including (infinitedimensional) Schrödinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual L2 costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions. [ABSTRACT FROM AUTHOR]

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations that assure the realizability of the optimized pulses. In this paper, we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient, and its slew rate, as well as constraints on the slice profile accuracy. For the latter, a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application, we reduce the power of refocusing pulses, which is important for spin echo-based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments. [ABSTRACT FROM PUBLISHER]

A class of infinite horizon optimal control problems involving nonsmooth cost functionals is discussed. The existence of optimal controls is studied for both the convex case and the nonconvex case, and the sparsity structure of the optimal controls promoted by the nonsmooth penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers. [ABSTRACT FROM AUTHOR]

Time optimal control problems for systems with impulsive controls are investigated. Sufficient conditions for the existence of time optimal controls are given. A dynamical programming principle is derived and Lipschitz continuity of an appropriately defined value functional is established. The value functional satisfies a Hamilton-Jacobi-Bellman equation in the viscosity sense. A numerical example for a rider-swing system is presented and it is shown that the reachable set is enlargered by allowing for impulsive controls, when compared to nonimpulsive controls. [ABSTRACT FROM AUTHOR]

Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc,L²(I)) is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem. [ABSTRACT FROM AUTHOR]

Optimal control problems in measure spaces governed by parabolic equations with are considered. The controls appear as spatial measure in the initial condition and as space-time measures as forcing functions. First order optimality conditions are derived and certain structural properties, in particular sparsity, are discussed. An framework for approximation if these highly irregular problems is also proposed. [ABSTRACT FROM AUTHOR]

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

An optimal control approach to a simplified reaction-diffusion system describing cardiac defibrillation is proposed that allows for joint optimization of shape and duration of defibrillation pulses. Within the framework, optimized multi-phasic pulses with low energy, short duration and/or low amplitude can be designed according to specific needs. The approach is based on a novel time optimal control formulation for the monodomain model, which takes into consideration the dynamical system properties of the uncontrolled equation. The highly complex dynamics requires a consistent discretization of first- and second-order information to guarantee effective optimization schemes leading to successful defibrillation. Numerical examples underline the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Optimal control techniques are investigated with the goal of terminating reentry waves in cardiac tissue models. In this computational study the Luo–Rudy phase-I ventricular action potential model is adopted which accounts for more biophysical details of cellular dynamics as compared to previously used phenomenological models. The parabolic and ordinary differential equations are solved as a coupled system and an AMG preconditioner is used to solve the discretized elliptic equation. The numerical results demonstrate that defibrillation is possible by delivering a single strong shock. The optimal control approach also leads to successful defibrillation and demands less total current. The present study motivates us to further investigate optimal control techniques on realistic geometries by incorporating the structural heterogeneity in the cardiac tissue. [ABSTRACT FROM AUTHOR]

A directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems. It allows for controls which are localized in space, where the spatial support is independent of time. Well-posedness of the optimal control problems is established and the optimality system is derived. It is used to establish structural properties of the minimizer. An a priori error analysis for finite element discretization is obtained, and numerical results illustrate the effects of the proposed cost functional and the convergence results. [ABSTRACT FROM AUTHOR]

Lp optimal control with p ∈ [0, 1) is investigated. The difficulty of natural lack of convexity and thus of weak lower semicontinuity is addressed by introducing appropriately chosen regularization terms. Existence results and necessary optimality conditions are obtained, and convergence of a monotone scheme is proved. Special attention is given to the particular case of optimal control problems with quadratic tracking and regularized L0 control costs are given. A maximum principle is derived and existence of controls, in some cases relaxed controls, is proved, and an estimate on the consequences of relaxation are estimated. [ABSTRACT FROM AUTHOR]

Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an L∞(Ω) regularity result for solutions of the first order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

A minimum effort optimal control problem for the undamped wave equation is considered which involves L-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Optimal control for an elliptic system with pointwise Euclidean norm constraints on the control variables is investigated. First order optimality conditions are derived in a manner that is amenable for numerical realisation. An efficient semismooth Newton algorithm is proposed based on this optimality system. Numerical examples are given to validate the superlinear convergence of the semismooth Newton algorithm. [ABSTRACT FROM AUTHOR]

This work proposes an optimal control approach for the termination of re-entry waves in cardiac electrophysiology. The control enters as an extracellular current density into the bidomain equations which are well established model equations in the literature to describe the electrical behavior of the cardiac tissue. The optimal control formulation is inspired, in part, by the dynamical systems behavior of the underlying system of differential equations. Existence of optimal controls is established and the optimality system is derived formally. The numerical realization is described in detail and numerical experiments, which demonstrate the capability of influencing and terminating reentry phenomena, are presented. [ABSTRACT FROM AUTHOR]

The semismooth Newton method for optimal control problems for systems of partial differential equations with polygonal constraints on the controls is developed. The Newton derivatives are characterized for the case of systems of dimension two, superlinear convergence is verified and a simple proof-of-concept numerical example is provided. [ABSTRACT FROM AUTHOR]

Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained. [ABSTRACT FROM AUTHOR]

An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived. [ABSTRACT FROM AUTHOR]

Time optimal control of the wave equation is analyzed on the basis of a regularized formulation which is considered as a bilevel optimization problem. For the lower level problems, which are constrained optimal control problems for the wave equation, a detailed sensitivity analysis is carried out. Further a semismooth Newton method is analyzed and proved to converge locally superlinearly. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

Optimal control problems in measure spaces lead to controls that have small support, which is desirable, e.g., in the context of optimal actuator placement. For problems governed by parabolic partial differential equations, well-posedness is guaranteed in the space of square-integrable measure-valued functions, which leads to controls with a spatial sparsity structure. A conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates. In particular, although the state is discretized, the control problem can still be formulated and solved in the measure space. Numerical examples illustrate the structural features of the optimal controls. [ABSTRACT FROM AUTHOR]

The problem of optimal placement of point sources is formulated as a distributed optimal control problem with sparsity constraints. For practical relevance, partial observations as well as partial and non-negative controls need to be considered. Although well-posedness of this problem requires a non-reflexive Banach space setting, a primal-predual formulation of the optimality system can be approximated well by a family of semi-smooth equations, which can be solved by a superlinearly convergent semi-smooth Newton method. Numerical examples indicate the feasibility for optimal light source placement problems in diffusive photochemotherapy. [ABSTRACT FROM AUTHOR]

Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions. Optimality conditions are derived and some of the structural properties of their solutions, in particular sparsity, are discussed. A framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates. [ABSTRACT FROM AUTHOR]

In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems. [ABSTRACT FROM PUBLISHER]

This work addresses an optimal control approach for a model problem in cardiac electrophysiology with the goal of extinction of a reentry phenomenon. After the introduction of the mathematical model, the derivation of the optimality system, the description of its discretization and a numerical feasibility study in a parallel environment are provided. [ABSTRACT FROM AUTHOR]

Optimal control for cardiac electrophysiology based on the bidomain equations in conjunction with the Fenton-Karma ionic model is considered. This generic ventricular model approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potentials. However, it is challenging due to the appearance of state-dependent discontinuities in the source terms. A computational framework for the numerical realization of optimal control problems is presented. Essential ingredients are a shape calculus based treatment of the sensitivities of the discontinuous source terms and a marching cubes algorithm to track iso-surface of excitation wavefronts. Numerical results exhibit successful defibrillation by applying an optimally controlled extracellular stimulus. [ABSTRACT FROM AUTHOR]