System Modeling and Optimization

This paper presents the modelling of the evolution of plasma equilibrium in the presence of external poloidal field circuits and passive structures. The optimization of plasma scenarios is formulated as an optimal control problem where the equations for the evolution of the plasma equilibrium are the constraints. The procedure determines the voltages applied to the external circuits that minimize a certain cost-function representing the distance to a desired plasma augmented by an energetic cost of the electrical system. A sequential quadratic programming method is used to solve the minimization of the cost-function and an application to the optimization of a discharge for ITER is shown.

We investigate the numerical approximation of solutions to some variational inequalities modeling the humid atmosphere when the saturation of water vapor in the air is taken into account. Here we describe part of our work [31] and extend our former results to the case where the saturation $$q_s$$ evolves with time.

A least-squares method is developed for estimating parameters in a size-structured population model with distributed states-at-birth from field data. First and second order finite difference schemes for approximating the nonlinear-nonlocal partial differential equation model are utilized in the least-squares problem. Convergence results for the computed parameters are established. Numerical results demonstrating the efficiency of the technique are provided.

In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.

We study the approximation of optimal control problems via the solution of a Hamilton-Jacobi equation in a tube around a reference trajectory which is first obtained solving a Model Predictive Control problem. The coupling between the two methods is introduced to improve the initial local solution and to reduce the computational complexity of the Dynamic Programming algorithm. We present some features of the method and show some results obtained via this technique showing that it can produce an improvement with respect to the two uncoupled methods.

The paper deals an investment timing problem appearing in real options theory. The present values from an investment project are modeled by general diffusion process. We find necessary and sufficient conditions under which the optimal investment time is induced by a threshold strategy. We study also conditions for optimality of the threshold strategy (over all threshold strategies) and discuss the connection between the solutions to the investment timing problem and the free-boundary problem.

This paper provides expressions for the boundary potential that provides the best electrostatic potential approximation of a given $$L^2$$ vector field on a nice bounded region in $${\mathbb R}^N$$. The permittivity of the region is assumed to be known and the potential is required to be zero on the conducting part of the boundary. The boundary potential is found by solving the minimization conditions and using a special basis of the trace space for the space of allowable potentials. The trace space is identified by its representation with respect to a basis of $$\varSigma $$-Steklov eigenfunctions.

We continue our efforts on modeling of the population dynamics of herbivorous insects in order to develop and implement effective pest control protocols. In the context of inverse problems, we explore the dynamic effects of pesticide treatments on Lygus hesperus, a common pest of cotton in the western United States. Fitting models to field data, we consider model selection for an appropriate mathematical model and corresponding statistical models, and use techniques to compare models. We address the question of whether data, as it is currently collected, can support time-dependent (as opposed to constant) parameter estimates.

In this work we introduce a novel operator $$\displaystyle \varDelta _{(p,q)}$$ as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear filtering scheme analogous to adaptive spatially dependent total variation and linear filtering. Well-posedness of the final parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator $$\displaystyle \varDelta _{(p,q)}$$.

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).

We deal with a regularized optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing behaviour of a viscoelastic plate vibrating against a rigid obstacle. A variable thickness of a plate plays the role of a control variable. The original problem for the deflection is regularized in order to have the uniqueness of a solution to the state problem and only the existence of an optimal thickness but also necessary optimality conditions.

In this paper, the abort landing problem is considered with reference to a point-mass aircraft model describing flight in a vertical plane. It is assumed that the pilot linearly increases the power setting to maximum upon sensing the presence of a windshear. This option is accounted for in the aircraft model and is not considered as a control. The only control is the angle of attack, which is assumed to lie between minimum and maximum values. The aim of this paper is to construct a feedback strategy that ensures a safe abort landing. An algorithm for solving nonlinear differential games is used for the design of such a strategy. The feedback strategy obtained is discontinuous in time and space so that realizations of control may have a bang-bang structure. To be realistic, outputs of the feedback strategy are being smoothed in time, and this signal is used as control.

This paper concerns the problem of aircraft control during the takeoff roll in the presence of severe wind gusts. It is assumed that the aircraft moves on the runway with a constant axial acceleration from a stationary position up to a specific speed at which the aircraft can go into flight. The lateral motion is controlled by the steering wheel and the rudder and affected by side wind. The aim of control is to prevent rolling out of the aircraft from the runway strip. Additionally, the lateral deviation, lateral speed, yaw angle, and yaw rate should remain in certain thresholds during the whole takeoff roll. The problem is stated as a differential game with state constraints. A grid method for computing the value function and optimal feedback strategies for the control and disturbance is used. The paper deals both with a nonlinear and linearized models of an aircraft on the ground. Simulations of the trajectories are presented.

The article discusses a numerical approach to solve optimal control problems in discrete time that involve continuous and discrete controls. Special attention is drawn to the modeling and treatment of dwell time constraints. For the solution of the optimal control problem in discrete time, a dynamic programming approach is employed. A numerical example is included that illustrates the impact of dwell time constraints in mixed integer optimal control.

In this paper we analyse an infimal convolution type regularisation functional called $$\mathrm {TVL}^{\infty }$$, based on the total variation ($$\mathrm {TV}$$) and the $$\mathrm {L}^{\infty }$$ norm of the gradient. The functional belongs to a more general family of $$\mathrm {TVL}^{p}$$ functionals ($$1<p\le \infty $$) introduced in [5]. There, the case $$1<p<\infty $$ is examined while here we focus on the $$p=\infty $$ case. We show via analytical and numerical results that the minimisation of the $$\mathrm {TVL}^{\infty }$$ functional promotes piecewise affine structures in the reconstructed images similar to the state of the art total generalised variation ($$\mathrm {TGV}$$) but improving upon preservation of hat–like structures. We also propose a spatially adapted version of our model that produces results comparable to $$\mathrm {TGV}$$ and allows space for further improvement.

In parameter estimation problems an important issue is the approximation of the confidence region of the estimated parameters. Especially for models based on differential equations, the needed computational costs require particular attention. For this reason, in many cases only linearized confidence regions are used. However, despite the low computational cost of the linearized confidence regions, their accuracy is often limited. To combine high accuracy and low computational costs, we have developed a method that uses only successive linearizations in the vicinity of an estimator. To accelerate the process, a principal axis decomposition of the covariance matrix of the parameters is employed. A numerical example illustrates the method.

Video streaming services need a server selection algorithm that allocates efficiently network and server resources. Solving this optimization problem requires information about current resources. A video streaming system that relies entirely on the service provider for this task needs an expensive monitoring infrastructure. In this paper, we consider a two-phase approach that reduces the monitoring requirements by involving the clients in the selection process: the provider recommends several servers based on limited information about the system’s resources, and the clients make the final decision, using information obtained by interacting with these servers. We implemented these selection methods in a simulator and compared their performance. The results show that the two-phase selection is effective, improving substantially the performance of lightweight service providers, with limited monitoring capabilities.

In this paper we formulate a time-optimal control problem in the space of probability measures endowed with the Wasserstein metric as a natural generalization of the correspondent classical problem in $${\mathbb {R}}^d$$ where the controlled dynamics is given by a differential inclusion. The main motivation is to model situations in which we have only a probabilistic knowledge of the initial state. In particular we prove first a Dynamic Programming Principle and then we give an Hamilton-Jacobi-Bellman equation in the space of probability measures which is solved by a generalization of the minimum time function in a suitable viscosity sense.

An optimal control problem of steady-state complex heat transfer with monotone objective functionals is under consideration. A coefficient function appearing in boundary conditions and reciprocally corresponding to the reflection index of the domain surface is considered as control. The concept of strong maximizing (resp. strong minimizing) optimal controls, i.e. controls that are optimal for all monotone objective functionals, is introduced. The existence of strong optimal controls is proven, and optimality conditions for such controls are derived. An iterative algorithm for computing strong optimal controls is proposed.

The main objective of this article is to present Bayesian optimal control over a class of non-autonomous linear stochastic discrete time systems with disturbances belonging to a family of the one parameter uniform distributions. It is proved that the Bayes control for the Pareto priors is the solution of a linear system of algebraic equations. For the case that this linear system is singular, we apply optimization techniques to gain the Bayesian optimal control. These results are extended to generalized linear stochastic systems of difference equations and provide the Bayesian optimal control for the case where the coefficients of these type of systems are non-square matrices. The paper extends the results of the authors developed for system with disturbances belonging to the exponential family.

The Hadamard semidifferential retains the advantages of the differential calculus such as the chain rule and semiconvex functions are Hadamard semidifferentiable. The semidifferential calculus extends to subsets of $${\mathbb {R}}^n$$ without Euclidean smooth structure. This set-up is an ideal tool to study the semidifferentiability of objective functions with respect to families of sets which are non-linear non-convex complete metric spaces. Shape derivatives are differentials for spaces endowed with Courant metrics. Topological derivatives are shown to be semidifferentials on the group of Lebesgue measurable characteristic functions.

We consider an optimal control problem with Volterra-type integral equations on a nonfixed time interval subject to endpoint constraints, mixed state-control constraints of equality and inequality type, and pure state inequality constraints. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We formulate first order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The presented conditions generalize the local maximum principle in optimal control problems with ordinary differential equations.

In the present paper a robust stabilization problem of continuous-time linear dynamic systems with Markov jumps and corrupted with multiplicative (state-dependent) white noise perturbations is considered. The robustness analysis is performed with respect to the intensity of the white noises. It is proved that the robustness radius depends on the solution of an algebraic system of coupled Lyapunov matrix equations.

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem.

The goal of this paper is to derive a traffic flow macroscopic model from a microscopic model with a transition function. At the microscopic scale, we consider a first order model of the form “follow the leader” i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider two different velocities and a transition zone. The transition zone represents a local perturbation operated by a Lipschitz function. After rescaling, we prove that the “cumulative distribution function” of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR model.

In this paper the well-posedness of some degenerate parabolic equations with a dynamic boundary condition is considered. To characterize the target degenerate parabolic equation from the Cahn–Hilliard system, the nonlinear term coming from the convex part of the double-well potential is chosen using a suitable maximal monotone graph. The main topic of this paper is the existence problem under an assumption for this maximal monotone graph for treating a wider class. The existence of a weak solution is proved.

We prove that the domain obtained by small perturbation of a Lipschtz domain is the union of a star-shaped domains with respect to every point of balls, such that the radius of the balls is independent of the perturbation. This result is useful in order to get uniform estimation for a fluid-structure interaction problem.

The present paper represents a continuation of [3]. There, we studied a new class of variational inequalities involving a pseudomonotone univalued operator and a multivalued operator, for which we obtained an existence result, among others. In the current paper we prove that this result remains valid under significantly weaker assumption on the multivalued operator. Then, we consider a new mathematical model which describes the equilibrium of an elastic body attached to a nonlinear spring on a part of its boundary. We use our abstract result to prove the weak solvability of this elastic model.

In the present paper we investigate nonlinear tracking problem under boundary control for the oscillation processes described by Fredholm integro-differential equations. When we investigate this problem we use notion of a weak generalized solution of the boundary value problem. Based on the maximum principle for distributed systems we obtain optimality conditions from which follow the nonlinear integral equation of optimal control and the differential inequality. We have developed an algorithm to construct the optimization problem solution. This solving method of a nonlinear tracking problem is constructive and can be used in applications.

We study the inverse Ingham type inequality for a wave equation in a ring. This leads to a conjecture on the zeros of Bessel cross product functions. We motivate the validity of the conjecture through numerical results. We do a complete analysis in the particular case of radial initial data, where an improved time of observability is available.

Various algorithmic differentiation tools have been developed and applied to large-scale simulation software for physical phenomena. Until now, two strictly disconnected approaches have been used to implement algorithmic differentiation (AD), namely, source transformation and operator overloading. This separation was motivated by different features of the programming languages such as Fortran and C++. In this work we have for the first time combined the two approaches to implement AD for C++ codes. Source transformation is used for core routines that are repetitive, where the transformed source can be optimized much better by modern compilers, and operator overloading is used to interconnect at the upper level, where source transformation is not possible because of complex language constructs of C++. We have also devised a method to apply the mixed approach in the same application semi-automatically. We demonstrate the benefit of this approach using some real-world applications.

Image restoration under sparsity constraints has received increased attention in recent years. This problem can be formulated as a nondifferentiable convex optimization problem whose solution is challenging. In this work, the non-differentiability of the objective is addressed by reformulating the image restoration problem as a nonnegatively constrained quadratic program which is then solved by a specialized Newton projection method where the search direction computation only requires matrix-vector operations. A comparative study with state-of-the-art methods is performed in order to illustrate the efficiency and effectiveness of the proposed approach.

The Hilbert Uniqueness Method introduced by J.-L. Lions in 1988 has great interest among scientists in the control theory, because it is a basic tool to get controllability results for evolutive systems. Our aim is to outline the Hilbert Uniqueness Method for first order coupled systems in the presence of memory terms in general Hilbert spaces. At the end of the paper we give some applications of our general results.

We consider a game problem of guaranteed positional control for a distributed system described by the phase field equations under incomplete information on system’s phase states. This problem is investigated from the viewpoint of the first player (the partner). For this player, a procedure for forming feedback controls is specified. This procedure is stable with respect to informational noises and computational errors and is based on the method of extremal shift and the method of stable sets from the theory of guaranteed positional control. It uses the idea of stable dynamical inversion of controlled systems.

In the present work we consider a frictional contact model with normal compliance. Firstly, we discuss the weak solvability of the model by means of two variational approaches. In a first approach the weak solution is a solution of a quasivariational inequality. In a second approach the weak solution is a solution of a mixed variational problem with solution-dependent set of Lagrange multipliers. Nextly, the paper focuses on the boundary optimal control of the model. Existence results, an optimality condition and some convergence results are presented.

The paper is concerned with the analysis and the numerical solution of the multimaterial topology optimization problems for bodies in unilateral contact. The contact phenomenon with Tresca friction is governed by the elliptic variational inequality. The structural optimization problem consists in finding such topology of the domain occupied by the body that the normal contact stress along the boundary of the body is minimized. The original cost functional is regularized using the multiphase volume constrained Ginzburg-Landau energy functional. The first order necessary optimality condition is formulated. The optimal topology is obtained as the steady state of the phase transition governed by the generalized Allen-Cahn equation. The optimization problem is solved numerically using the operator splitting approach combined with the projection gradient method. Numerical examples are provided and discussed.

We discuss recent constructive parametrizations approaches for implicit systems, via systems of ordinary differential equations. We also present the notion of generalized solution, in the critical case and indicate some numerical examples in dimension two and three, using MatLab. In shape optimizations problems, using this method, we introduce general optimal control formulations in the boundary observation case. This extends previous work of the authors on optimal design problems with distributed cost functional.

We study finite element approximations of Riesz representatives of shape gradients. First, we provide a general perspective on its error analysis. Then, we focus on shape functionals constrained by elliptic boundary value problems and $$H^1$$-representatives of shape gradients. We prove linear convergence in the energy norm for linear Lagrangian finite element approximations. This theoretical result is confirmed by several numerical experiments.

The paper discusses a class of bilevel optimal control problems with optimal control problems at both levels. The problem will be transformed to an equivalent single level problem using the value function of the lower level optimal control problem. Although the computation of the value function is difficult in general, we present a pursuit-evasion Stackelberg game for which the value function of the lower level problem can be derived even analytically. A direct discretization method is then used to solve the transformed single level optimal control problem together with some smoothing of the value function.

We consider a system of rigid bodies subjected to unilateral constraints with soft contact and dry friction. When the constraints are saturated, velocity jumps may occur and the dynamics is described in generalized coordinates by a second-order measure differential inclusion for the unknown configurations. Observing that the right velocity obeys a minimization principle, a time-stepping algorithm is proposed. It allows to construct a sequence of approximate solutions satisfying at each time-step a discrete contact law which mimics the behaviour of the system in case of collision. In case of tangential contact, dry friction may lead to indeterminacies such as the famous Painlevé’s paradoxes. By a precise study of the asymptotic properties of the scheme, it is shown that the limit of the approximate trajectories exhibits the same kind of indeterminacies.

This paper investigates an optimal control problem associated with a complex nonlinear system of multiple delay differential equations modeling the development of healthy and leukemic cell populations incorporating the immune system. The model takes into account space competition between normal cells and leukemic cells at two phases of the development of hematopoietic cells. The control problem consists in optimizing the treatment effect while minimizing the side effects. The Pontryagin minimum principle is applied and important conclusions about the character of the optimal therapy strategy are drawn.

We consider a mathematical model which describes the equilibrium of an electro-elastic beam in contact with an electrically conductive foundation. The model is constructed by coupling the beam equation with the one dimensional piezoelectricity system obtained in [13]. We state the unique weak solvability of the model as well as the continuous dependence of the weak solution with respect to the data. We also introduce a discrete scheme for which we perform the numerical analysis, including convergence and error estimates results. Finally, we present numerical simulations in the study of a test problem.

We present two models for engineering processes, where thermal effects and time-dependent domains play an important role. Typically, the parabolic heat equation is coupled with other equations. Challenges for the optimization of such systems are presented.The first model describes a milling process, where material is removed and heat is produced by the cutting, leading to thermomechanical distortion. Goal is the minimization of these distortions.The second model describes the melting and solidification of metal, where the geometry is a result of free-surface flow of the liquid and the microstructure of the re-solidified material is important for the quality of the produced preform.

A boundary value problem with state constraints is under consideration for a nonlinear noncoercive Hamilton-Jacobi equation. The problem arises in molecular biology for the Crow – Kimura model of genetic evolution. A new notion of continuous generalized solution to the problem is suggested. Connections with viscosity and minimax generalized solutions are discussed. In this paper the problem is studied for the case of additional requirements to structure of solutions. Constructions of the solutions with prescribed properties are provided and justified via dynamic programming and calculus of variations. Results of simulations are exposed.

Perturbed inverse reconstruction problems for controlled dynamic systems are under consideration. A sample history of the actual trajectory is known. This trajectory is generated by a control, which isn’t known. Moreover, the deviation of the samples from the actual trajectory satisfies the known estimate of the sample error. The inverse problem with perturbed (inaccurate) sample of trajectory consists of reconstructing trajectories which are close to the actual trajectory in C. Controls generating the trajectories should be close in $$L_{2}$$ to the normal control generating the actual trajectory and have the least norm in $$L_{2}$$. A numerical method for solving this problem is suggested. The application of the suggested method is illustrated by the graphics.

This article is devoted to studying dual regularization method as applied to parametric convex optimal control problem of controlled third boundary-value problem for parabolic equation with boundary control and with equality and inequality pointwise state constraints. These constraints are understood as ones in the Hilbert space $$L_2$$. A major advantage of the constraints of the original problem which are understood as ones in $$L_2$$ is that the resulting dual regularization algorithm is stable with respect to errors in the input data and leads to the construction of a minimizing approximate solution in the sense of J. Warga. Simultaneously, this dual algorithm yields the corresponding necessary and sufficient conditions for minimizing sequences, namely, the stable, with respect to perturbation of input data, sequential or, in other words, regularized Lagrange principle in nondifferential form and Pontryagin maximum principle for the original problem. Regardless of the fact that the stability or instability of the original optimal control problem, they stably generate a minimizing approximate solutions for it. For this reason, we can interpret these regularized Lagrange principle and Pontryagin maximum principle as tools for direct solving unstable optimal control problems and reducing to them unstable inverse problems.

The convergence of a family of continuous distributed mixed elliptic optimal control problems ($$ P_{\alpha } $$), governed by elliptic variational equalities, when the parameter $$ \alpha \to \infty $$ was studied in Gariboldi - Tarzia, Appl. Math. Optim., 47 (2003), 213-230 and it has been proved that it is convergent to a distributed mixed elliptic optimal control problem ($$ P $$). We consider the discrete approximations ($$ P_{h\alpha } $$) and ($$ P_{h} $$) of the optimal control problems ($$ P_{\alpha } $$) and ($$ P $$) respectively, for each $$ h > 0 $$ and $$ \alpha > 0 $$. We study the convergence of the discrete distributed optimal control problems ($$ P_{h\alpha } $$) and ($$ P_{h} $$) when $$ h \to 0 $$, $$ \alpha \to \infty $$ and $$ (h,\alpha ) \to (0, +\infty ) $$ obtaining a complete commutative diagram, including the diagonal convergence, which relates the continuous and discrete distributed mixed elliptic optimal control problems $$ \left( {P_{h\alpha } } \right),\;\left( {P_{\alpha } } \right),\;\left( {P_{h} } \right) $$ and ($$ P $$) by taking the corresponding limits. The convergent corresponds to the optimal control, and the system and adjoint system states in adequate functional spaces.

We consider a particular model for electromagnetic fields in the context of optimal control. Special emphasis is laid on a non-standard H-based formulation of the equations of low-frequency electromagnetism in multiply connected conductors. By this technique, the low-frequency Maxwell equations can be solved with reduced computational complexity. We show the well-posedness of the system and derive the sensitivity analysis for different models of controls.

The aim of this work is to test the Levemberg Marquardt and BFGS (Broyden Fletcher Goldfarb Shanno) algorithms, implemented by the matlab functions lsqnonlin and fminunc of the Optimization Toolbox, for modeling the kinetic terms occurring in chemical processes of adsorption. We are interested in tests with noisy data that are obtained by adding Gaussian random noise to the solution of a model with known parameters. While both methods are very precise with noiseless data, by adding noise the quality of the results is greatly worsened. The semi-convergent behaviour of the relative error curves is observed for both methods. Therefore a stopping criterion, based on the Discrepancy Principle is proposed and tested. Great improvement is obtained for both methods, making it possible to compute stable solutions also for noisy data.