System Modeling and Optimization

In the second order analysis of infinite dimension optimization problems, we have to deal with the so-called two-norm discrepancy. As a consequence of this fact, the second order optimality conditions usually imply local optimality in the

L

 ∞ 

sense. However, we have observed that the

L

2

local optimality can be proved for many control problems of partial differential equations. This can be deduced from the standard second order conditions. To this end, we make some quite realistic assumptions on the second derivative of the cost functional. These assumptions do not hold if the control does not appear explicitly in the cost functional. In this case, the optimal control is usually of bang-bang type. For this type of problems we also formulate some new second order optimality conditions that lead to the strict

L

2

local optimality of the bang-bang controls.

In order to achieve prescribed drug release kinetics over long therapeutic periods, bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers are currently envisioned. The modelling of drug release in the presence of degradation of the polymer matrix and surface erosion is quite complex. Yet, simple reliable mathematical models validated against experimental data are now available to help in classifying neat polymers and in predicting the release dynamics from polymer blends. In this paper, we survey a two-parameter quadratic ODE model that has been validated against experimental data for the release of paclitaxel from a broad range of biodegradable polymers and a quadratic semi-permeable membrane PDE model that mimics the ODE model and could readily be extended to drug eluding stents.

The solution of chance constrained optimization problems by means of empirical approximation of the underlying multivariate distribution has recently become a popular alternative to conventional methods due to the efficient application of appropriate mixed integer programming techniques. As the complexity of required computations depends on the sample size used for approximation, exponential estimates for the precision of optimal solutions or optimal values have become a key argument for controlling the sample size. However, these exponential estimates may involve unknown constants such that the required sample size to approximate the solution of a problem may become arbitrarily large. We will illustrate this effect for Gaussian distributions.

In this paper we provide a convergence rates result for a modified version of Landweber iteration with a priori regularization parameter choice in a Banach space setting.

In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces.

We investigate adaptive methods for optimal control problems with finitely many control parameters. We analyze a-posteriori error estimates based on verification of second-order sufficient optimality conditions. Reliability and efficiency of the error estimator is shown. The estimator is used in numerical tests to guide adaptive mesh refinement.

We deal with initial-boundary value problems describing vertical vibrations of viscoelastic von Kármán-Donnell shells with a rigid inner obstacle. The short memory (Kelvin-Voigt) material is considered. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the penalization method.

I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term

g

through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over

g

for each parameter

h

 > 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot’s inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter

h

. We also prove, when

h

 → + ∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition.

II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems.

III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter

h

 > 0, but in this case the convergence when

h

 → + ∞ is still an open problem.

Recently a new class of differential variational inequalities has been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. In this short note we lift this class of nonsmooth dynamical systems to the level of a Hilbert space, but focus to linear input/output systems. This covers in particular linear complementarity systems where the underlying convex constraint set in the variational inequality is specialized to an ordering cone.

The purpose of this note is two-fold. Firstly, we provide an existence result based on maximal monotone operator theory. Secondly we are concerned with stability of the solution set of linear differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated linear maps and the constraint set.

We propose a model order reduction (MOR) approach for networks containing simple and complex components. Simple components are modeled by linear ODE (and/or DAE) systems, while complex components are modeled by nonlinear PDE (and/or PDAE) systems. These systems are coupled through the network topology using the Kirchhoff laws. As application we consider MOR for electrical networks, where semiconductors form the complex components which are modeled by the transient drift-diffusion equations (DDEs). We sketch how proper orthogonal decomposition (POD) combined with discrete empirical interpolation (DEIM) and passivity-preserving balanced truncation methods for electrical circuits (PABTEC) can be used to reduce the dimension of the model. Furthermore we investigate residual-based sampling to construct reduced order models which are valid over a certain parameter range.

An optimal control problem to find the fastest collision-free trajectory of a robot is presented. The dynamics of the robot is governed by ordinary differential equations. The collision avoidance criterion is a consequence of Farkas’s lemma and is included in the model as state constraints. Finally an active set strategy based on backface culling is added to the sequential quadratic programming which solves the optimal control problem.

We discuss a technical approach, based on the method of regularized extremal shift (RES), intended to help solve problems of stable control of uncertain dynamical systems. Our goal is to demonstrate the essence and abilities of the RES technique; for this purpose we construct feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed system described by a parabolic equation. The controller is “resource-saving” in a sense that control resource spent for approximate tracking do not exceed those needed for tracking in an “ideal” situation where the current values of the input disturbance are fully observable.

In the paper we derive new necessary optimality conditions for optimal control of differential equations systems with discontinuous right hand side. The main attention is paid to a situation when an optimal trajectory slides on the discontinuity surface. The new conditions, derived in the paper, are essential and do not follow from any known necessary conditions for such systems.

We are interested in a class of numerical schemes for the optimization of nonlinear hyperbolic partial differential equations. We present continuous and discretized relaxation schemes for scalar, one– conservation laws. We present numerical results on tracking typew problems with nonsmooth desired states and convergence results for higher–order spatial and temporal discretization schemes.

We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient.

Due to their frequently observed lack of convexity and/or smoothness, stochastic programs with joint probabilistic constraints have been considered as a hard type of constrained optimization problems, which are rather demanding both from the computational and robustness point of view. Dependence of the set of solutions on the probability distribution rules out the straightforward construction of the convexity-based global contamination bounds for the optimal value; at least local results for probabilistic programs of a special structure will be derived. Several alternative approaches to output analysis will be mentioned.

The paper deals with polyhedral estimates for reachable tubes of differential systems with a multiplicative uncertainty, namely linear systems with set-valued uncertainties in initial states, additive inputs and coefficients of the system. We present nonlinear parametrized systems of ordinary differential equations (ODE) which describe the evolution of the parallelotope-valued estimates for reachable sets (time cross-sections of the reachable tubes). The main results are obtained for internal estimates. In fact, a whole family of the internal estimates is introduced. The properties of the obtained ODE systems (such as existence and uniqueness of solutions, nondegeneracy of estimates) are investigated. Using some optimization procedure we also obtain a differential inclusion which provides nondegenerate internal estimates. Examples of numerically constructed external and internal estimates are presented.

An algorithm for solving quadratic, two-stage stochastic problems is developed. The algorithm is based on the framework of the Branch and Fix Coordination (BFC) method. These problems have continuous and binary variables in the first stage and only continuous variables in the second one. The objective function is quadratic and the constraints are linear. The nonanticipativity constraints are fulfilled by means of the twin node family strategy. On the basis of the BFC method for two-stage stochastic linear problems with binary variables in the first stage, an algorithm to solve these stochastic quadratic problems is designed. In order to gain computational efficiency, we use scenario clusters and propose to use either outer linear approximations or (if possible) perspective cuts. This algorithm is implemented in C++ with the help of the Cplex library to solve the quadratic subproblems. Numerical results are reported.

We consider a stochastic control problem of beating a stochastic benchmark. The problem is considered in an incomplete market setting with external economic factors. The investor preferences are modelled in terms of HARA-type utility functions and trading takes place in a finite time horizon. The objective of the investor is to minimize his expected loss from the outperformance of the benchmark compared to the portfolio terminal wealth, and to specify the optimal investment strategy. We prove that for considered loss functions the corresponding Bellman equation possesses a unique solution. This solution guaranties the existence of a well defined investment strategy. We prove also under which conditions the verification theorem for the obtained solution of the Bellman equation holds.

This paper examines the objective of optimally harvesting a single species in a stochastic environment. This problem has previously been analyzed in [1] using dynamic programming techniques and, due to the natural payoff structure of the price rate function (the price decreases as the population increases), no optimal harvesting policy exists. This paper establishes a relaxed formulation of the harvesting model in such a manner that existence of an optimal relaxed harvesting policy can not only be proven but also identified. The analysis imbeds the harvesting problem in an infinite-dimensional linear program over a space of occupation measures in which the initial position enters as a parameter and then analyzes an auxiliary problem having fewer constraints. In this manner upper bounds are determined for the optimal value (with the given initial position); these bounds depend on the relation of the initial population size to a specific target size. The more interesting case occurs when the initial population exceeds this target size; a new argument is required to obtain a sharp upper bound. Though the initial population size only enters as a parameter, the value is determined in a closed-form functional expression of this parameter.

A change of shares of credits portfolio is described by Markov chain with discrete time. A credit state is determined on as an accessory to some group of credits depending on presence of indebtedness and its terms. We use a model with discrete time and fix the system state through identical time intervals - once a month. It is obvious that the matrix of transitive probabilities is known incompletely. Various approaches to the matrix estimation are studied and methods of forecast the portfolio risk are proposed. The portfolio risk is set as a share of problematic loans. We propose a method to calculate necessary reserves on the base of the considered model.

We provide a theoretical framework for model predictive control of infinite-dimensional systems, like, e.g., nonlinear parabolic PDEs, including stochastic disturbances of the input signal, the output measurements, as well as initial states. The necessary theory for implementing the MPC step based on an LQG design for infinite-dimensional linear time-invariant systems is presented. We also briefly discuss the necessary ingredients for the numerical computations using the derived theory.

We discuss a problem of the dynamic reconstruction of unknown input controls in nonlinear vector equations. A regularizing algorithm is proposed for reconstructing these controls simultaneously with the processes. The algorithm is stable with respect to informational noises and computational errors.

Finite-difference schemes for the computation of value functions of nonlinear differential games with non-terminal payoff functional and state constraints are proposed. The solution method is based on the fact that the value function is a generalized viscosity solution of the corresponding Hamilton-Jacobi-Bellman-Isaacs equation. Such a viscosity solution is defined as a function satisfying differential inequalities introduced by M. G. Crandall and P. L. Lions. The difference with the classical case is that these inequalities hold on an unknown in advance subset of the state space. The convergence rate of the numerical schemes is given. Numerical solution to a non-trivial three-dimensional example is presented.

Continuing research in [13] and [14] on well-posedness of the optimal time control problem with a constant convex dynamics in a Hilbert space we adapt one of the regularity conditions obtained there to a slightly more general problem, where nonaffine additive term appears. We prove existence and uniqueness of a minimizer in this problem as well as continuous differentiability of the value function, which can be seen as the viscosity solution to a Hamilton-Jacobi equation, near the boundary.

We consider the subcritical gas flow through star-shaped pipe networks. The gas flow is modeled by the isothermal Euler equations with friction. We stabilize the isothermal Euler equations locally around a given stationary state on a finite time interval. For the stabilization we apply boundary feedback controls with time-varying delays. The delays are given by

C

1

-functions with bounded derivatives. In order to analyze the system evolution, we introduce an

L

2

-Lyapunov function with delay terms. The boundary controls guarantee the exponential decay of the Lyapunov function with time.

The control of complex forming processes (e.g., glass forming processes) is a challenging topic due to the mostly strongly nonlinear behavior and the spatially distributed nature of the process. In this paper a new approach for the real-time control of a spatially distributed temperature profile of an industrial glass forming process is presented. As the temperature in the forming zone cannot be measured directly, it is estimated by the numerical solution of the partial differential equation for heat transfer by a finite element scheme. The numerical solution of the optimization problem is performed by the solver HQP (Huge Quadratic Programming). In order to meet real-time requirements, in each sampling interval the full finite element discretization of the temperature profile is reduced considerably by a spline approximation. Results of the NMPC concept are compared with conventional PI control results. It is shown that NMPC stabilizes the temperature of the forming zone much better than PI control. The proposed NMPC scheme is robust against model mismatch of the disturbance model. Furthermore, the allowed parameter settings for a real-time application (i.e., control horizon, sampling period) have been determined. The approach can easily be adapted to other forming processes where the temperature profile shall be controlled.

We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and that acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a

C

0

 −semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. To establish this result, we introduce a suitable energy function and use multiplier method together with an estimate taken from [3] (Lemma 7.2) and compactness-uniqueness arguments.

In optimal control problems with infinite time horizon, arising in models of economic growth, there are essential difficulties in analytical and even in numerical construction of solutions of Hamiltonian systems. The problem is in stiff properties of differential equations of the maximum principle and in non-stable character of equilibrium points connected with corresponding transversality conditions. However, if a steady state exists and meets several conditions of regularity then it is possible to construct a nonlinear stabilizer for the Hamiltonian system. This stabilizer inherits properties of the maximum principle, generates a nonlinear system with excluded adjoint variables and leads its trajectories to the steady state. Basing on the qualitative theory of differential equations, it is possible to prove that trajectories generated by the nonlinear stabilizer are close to solutions of the original Hamiltonian system, at least locally, in a neighborhood of the steady state. This analysis allows to create stable algorithms for construction of optimal solutions.

The tracking control design for setpoint transitions of a quasi-linear diffusion-convection-reaction system with boundary control is considered. For this a suitable model-based feedforward control is determined that relies on the flatness-based parametrization of the control input. A receding horizon feedback control is added within a two-degrees-of-freedom control scheme to account for disturbances, model inaccuracies, and input constraints.The tracking performance of this control scheme is shown by means of simulation studies.

Time-varying discrete-time linear systems may be temporarily uncontrollable and unreconstructable. This is vital knowledge to both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability requires considering discrete-time systems with variable dimensions and the j-step, k-step Kalman decomposition. In this note for linear discrete-time systems with variable dimensions measures of temporal and one-step stabilizability and detectability are developed. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed loop system when designing a static state or dynamic output feedback controller. The measures are calculated by solving specific linear quadratic cheap control problems.

While active flow control is an established method for controlling flow separation on vehicles and airfoils, the design of the actuation is often done by trial and error. In this paper, the development of a discrete and a continuous adjoint flow solver for the optimal control of unsteady turbulent flows governed by the incompressible Reynolds-averaged Navier-Stokes equations is presented. Both approaches are applied to testcases featuring active flow control of the blowing and suction type and are compared in terms of accuracy of the computed gradient.

We deal with well-posedness and asymptotic dynamics of a class of coupled systems consisting of linearized 3D Navier–Stokes equations in a bounded domain and a classical (nonlinear) elastic plate/shell equation. We consider three models for plate/shell oscillations: (a) the model which accounts for transversal displacement of a flexible flat part of the boundary only, (b) the model for in-plane motions of a flexible flat part of the boundary, (c) the model which accounts for both transversal and longitudinal displacements. For all three cases we present well-posedness results and prove existence of a compact global attractor. In the first two cases the attractor is of finite dimension and possesses additional smoothness. We do not assume any kind of mechanical damping in the plate component in the case of models (a) and (b). Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system in the latter cases.

The semilinear normal parabolic equations corresponding to 3D Navier-Stokes system have been derived. The explicit formula for solution of normal parabolic equations with periodic boundary conditions has been obtained. It was shown that phase space of corresponding dynamical system consists of the set of stability (where solutions tends to zero as time

t

 → ∞), the set of explosions (where solutions blow up during finite time) and intermediate set. Exact description of these sets has been given.

We present a nonlinear model predictive framework for closed-loop control of two-phase flows governed by the Cahn-Hilliard Navier-Stokes system. We adapt the concept for instantaneous control from [6,12,16] to construct distributed closed-loop control strategies for two-phase flows. It is well known that distributed instantaneous control is able to stabilize the Burger’s equation [16] and also the Navier-Stokes system [6,12]. In the present work we provide numerical investigations which indicate that distributed instantaneous control also is well suited to stabilize the Cahn-Hilliard Navier-Stokes system.

We present a weak formulation for a steady fluid-structure interaction problem using an embedding domain technique with penalization. Except of the penalizing term, the coefficients of the fluid problem are constant and independent of the deformation of the structure, which represents an advantage of this approach. A second advantage of this model is the fact that the continuity of the stress at the fluid-structure interface does not appear explicitly. Numerical results are presented.

In this note a concept of

ε

-level set function is introduced, i.e. a function which approximates a level set function satisfying the Hamilton-Jacobi inequality. We prove that each Lipschitz continuous solution of the Hamilton-Jacobi inequality is an

ε

-level set function. Next, a numerical approximation of the level set function is presented, i.e. method for the construction of an

ε

-level set function.

The paper aims to illustrate the algorithm developed in the paper [6] in some specific problems of shape optimization issued from fluid mechanics. Using the fictitious domain method with penalization, the fluid equations will be solved in a fixed domain. The admissible shapes are parametrized by continuous function defined in the fixed domain, then the shape optimization problem becomes an optimal control problem, where the control is the parametrization of the shape. We get the directional derivative of the cost function by solving co-state equation. Numerical results are obtained using a gradient type algorithm.

We present a novel, thermodynamically consistent, model for the charged-fluid flow and the deformation of the morphology of polymer electrolyte membranes (PEM) in hydrogen fuel cells. The solid membrane is assumed to obey linear elasticity, while the pore is completely filled with protonated water, considered as a Stokes flow. The model comprises a system of partial differential equations and boundary conditions including a free boundary between liquid and solid. Our problem generalizes the well-known Nernst-Planck-Poisson-Stokes system by including mechanics. We solve the coupled non-linear equations numerically and examine the equilibrium pore shape. This computationally challenging problem is important in order to better understand material properties of PEM and, hence, the design of hydrogen fuel cells.

This work deals with the development and application of reduction strategies for

real-time

and

many query

problems arising in fluid dynamics, such as shape optimization, shape registration (reconstruction), and shape parametrization. The proposed strategy is based on the coupling between reduced basis methods for the reduction of computational complexity and suitable shape parametrizations – such as free-form deformations or radial basis functions – for low-dimensional geometrical description. Our focus is on problems arising in haemodynamics: efficient shape parametrization of cardiovascular geometries (e.g. bypass grafts, carotid artery bifurcation, stenosed artery sections) for the rapid blood flow simulation – and related output evaluation – in domains of variable shape (e.g. vessels in presence of growing stenosis) provide an example of a class of problems which can be recast in the

real-time

or in the

many-query

context.

The paper deals with the shape and topology optimization of the elliptic variational inequalities using the level set approach. The standard level set method is based on the description of the domain boundary as an isocountour of a scalar function of a higher dimensionality. The evolution of this boundary is governed by Hamilton-Jacobi equation. In the paper a piecewise constant level set method is used to represent interfaces rather than the standard method. The piecewise constant level set function takes distinct constant values in each subdomain of a whole design domain. Using a two-phase approximation and a piecewise constant level set approach the original structural optimization problem is reformulated as an equivalent constrained optimization problem in terms of the level set function. Necessary optimality condition is formulated. Numerical examples are provided and discussed.

In this paper we describe a novel framework for finding numerical solutions to a wide range of shape optimization problems. It is based on classical dynamic programming approach augmented with discretization of the space of trajectories and controls. This allows for straightforward algorithmic implementation. This method has been used to solve a well known problem called the ”dividing tube problem”, a state problem related to fluid mechanics, that requires simultaneous topology and shape optimization in case of elastic contact problems and involves solving the Navier-Stokes equations for viscous incompressible fluids.

We study the shape differentiability of a cost function for the steady flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.

For shape optimization problems associated to stationary Navier-Stokes equations, we introduce the corresponding finite element approximation and we prove convergence results.

The paper provides shape derivative analysis for the wave equation with mixed boundary conditions on a moving domain Ω

s

in the case of non smooth neumann boundary datum. The key ideas in the paper are (i) bypassing the classical sensitivity analysis of the state by using parameter differentiability of a functional expressed in the form of Min-Max of a convex-concave Lagrangian with saddle point, and (ii) using a new regularity result on the solution of the wave problem (where the Dirichlet condition on the fixed part of the boundary is essential) to analyze the strong derivative.

The exactness of the penalization for the exact

l

1

penalty function method used for solving nonsmooth constrained optimization problems with both inequality and equality constraints is considered. Thus, the equivalence between the sets of optimal solutions in the nonsmooth constrained optimization problem and its associated penalized optimization problem with the exact

l

1

penalty function is established under locally Lipschitz invexity assumptions imposed on the involved functions.

One of the most important factors of users comfort inside building is air temperature. From the other side one of the most biggest position in home budget is price for heat. Mutually exclusive indices are the cause that the control of temperature task using the smallest amount of energy as it is possible is very difficult. In this paper is presented simple model of temperature changes inside building base on lumped capacity method. Using this method finally obtains mathematical model of temperature changes which model is equivalent in structure to electrical RC- network. The model is composed of linear differential equations. Based on this mathematical model the simple algorithm controlling of temperature inside room is proposed. In this article are also included numerical simulations of the proposed solutions.

This paper presents the application of the

Discrete Linear Quadratic Control (DLQC)

method for a parameter optimization problem in a marine ecosystem model. The ecosystem model simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column with temperature and turbulent diffusivity profiles taken from a three-dimensional ocean circulation model. We present the linearization method which is based on the available observations. The linearization is necessary to apply the DLQC method on the nonlinear system of state equations. We show the form of the linearized time-variant problems and the resulting two algebraic Riccati Equations. By using the DLQC method, we are able to introduce temporally varying periodic model parameters and to significantly improve – compared to the use of constant parameters – the fit of the model output to given observational data.

The article suggests a conceptual model-based simulation method with the aim to detect collision of cars in all-day road traffic. The benefit of the method within a driver assistance system would be twofold. Firstly, unavoidable accidents could be detected and appropriate actions like full braking maneuvers could be initiated in due course. Secondly, in case of an avoidable accident the algorithm is able to suggest an evasion trajectory that could be tracked by a future active steering driver assistance system. The algorithm exploits numerical optimal control techniques and reachable set analysis. A parametric sensitivity analysis is employed to investigate the influence of inaccurate sensor measurements.

We present the investigation of a biogeochemical marine ecosystem model used as part of the climate change research focusing on the enhanced carbon dioxid concentration in the atmosphere. Numerical parameter optimization has been performed to improve represention of observational data using data assimilation techniques. Several local minima were found but no global optimum could be identified. To detect the actual capability of the model in simulating natural systems, a theoretical analysis of the model equations is conducted. Here, basic properties such as continuity and positivity of the model equations are investigated.

The electric market regulation in Spain (MIBEL) establishes the rules for bilateral and futures contracts in the day-ahead optimal bid problem. Our model allows a price-taker generation company to decide the unit commitment of the thermal units, the economic dispatch of the bilateral and futures contracts between the thermal units and the optimal sale bids for the thermal units observing the MIBEL regulation. The uncertainty of the spot prices is represented through scenario sets. We solve this model on the framework of the Branch and Fix Coordination metodology as a quadratic two-stage stochastic problem. In order to gain computational efficiency, we use scenario clusters and propose to use perspective cuts. Numerical results are reported.

We study a nonlinear transmission problem for a plate which consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. Main result is the proof of asymptotic smoothness of this dynamical system and existence of a compact global attractor in special cases.

This paper is devoted to singular calculus of variations problems with constraints which are not regular mappings at the solution point, e.i. its derivatives are not surjective. We pursue an approach based on the constructions of the

p

-regularity theory. For

p

-regular calculus of variations problem we present necessary conditions for optimality in singular case and illustrate our results by classical example of calculus of variations problem.

The paper presents mathematical and implementation challenges associated with testing of embedded software systems with dynamic behavior. These challenges are related to notation of tests, calculation of test coverage, implementation of a test comparator, and automatic generation of test cases. Some author’s ideas and solutions are presented with the help of abstract models that describe behavior of the software systems. The models are represented using the state space (or input/state/output) notation. An application example is given to illustrate theoretical analysis and mathematical formulation.

In this work we present an iterative algorithm for solving a parameter identification problem relative to a system of diffusion, convection and reaction equations. The parameters to estimate are the retardation factors, diffusivity, reaction and transport coefficients relative to a model of pollutant transport with chemical reaction. The proposed method solves the nonlinear least squares problem by means of a sequence of constrained optimization problems. The algorithm does not depend on the type of discretization method used to solve the state equation. The results reported in the numerical tests show the efficiency of the algorithm in terms of performance and solution quality.

We propose a variational formulation of a macroscopic model for crowd motion involving a conservation law describing mass conservation coupled with an eikonal equation giving the flow direction. To get a self contain paper we recall many results concerning flow mapping and convection process associated with non smooth vector field

V

.