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    Tytuł:
    Second-order optimality conditions in nonsmooth vector optimization
    Autorzy:
    Yadav, Priyanka
    Powiązania:
    https://bibliotekanauki.pl/articles/31342762.pdf
    Data publikacji:
    2023
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    vector optimization
    cones
    nonsmooth second-order cone-convexity
    second-order optimality
    duality
    Opis:
    In this paper, we introduce new classes of nonsmooth second-order cone-convex functions and respective generalizations in terms of first and second-order directional derivative. These classes encapsulate several already existing classes of cone-convex functions and their weaker variants. Second-order KKT type sufficient optimality conditions and duality results for a nonsmooth vector optimization problem are proved using these functions. The results have been supported by examples.
    Źródło:
    Control and Cybernetics; 2023, 52, 1; 35-51
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second order optimality conditions for bang-bang control problems
    Autorzy:
    Maurer, H.
    Osmolovskii, N. P.
    Powiązania:
    https://bibliotekanauki.pl/articles/970524.pdf
    Data publikacji:
    2003
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    sterowanie przekaźnikowe
    bang-bang control
    second order necessary and sufficient conditions
    critical cone
    transformation of quadratic forms
    numerical verification of second order conditions
    van der Pol oscillator
    Opis:
    Second order necessary and sufficient optimality conditions for bang-bang control problems have been studied in Milyutin, Osmolovskii (1998). These conditions amount to testing the positive (semi-)definiteness of a quadratic form on a critical cone. The assumptions are appropriate for numerical verification only in some special cases. In this paper, we study various transformations of the quadratic form and the critical cone which will be tailored to different types of control problems in practice. In particular, by means of a solution to a linear matrix differential equation, the quadratic form can be converted to perfect squares. We demonstrate by three practical examples that the conditions obtained can be verified numerically.
    Źródło:
    Control and Cybernetics; 2003, 32, 3; 555-584
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results
    Autorzy:
    Osmolovskii, N. P.
    Maurer, H.
    Powiązania:
    https://bibliotekanauki.pl/articles/970559.pdf
    Data publikacji:
    2005
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    sterowanie przekaźnikowe
    stożek krytyczny
    forma kwadratowa i równoważność
    bang-bang control
    second order necessary and sufficient conditions
    critical cone
    Opis:
    Second order optimality conditions have been derived in the literature in two different forms. Osmolovskii (1988a, 1995, 2000, 2004) obtained second order necessary and sufficient conditions requiring that, a certain quadratic form be positive (semi)-definite on a critical cone. Agrachev, Stefani, Zezza (2002) first, reduced the bang-bang control problem to finite-dimensional optimization and then show that well-known sufficient optimality conditions for this optimization problem supplemented by the strict bang-bang property furnish sufficient conditions for the bang-bang control problem. In this paper, we establish the equivalence of both forms of sufficient conditions and give explicit relations between corresponding Lagrange multipliers and elements of critical cones. Part 1 summarizes the main results while detailed proofs will be given in Part 2.
    Źródło:
    Control and Cybernetics; 2005, 34, 3; 927-950
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second-order optimality conditions for semilinear elliptic control problems with constraints on the gradient of the state
    Autorzy:
    Casas, E.
    Fernandez, J.
    Mateos, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/206719.pdf
    Data publikacji:
    1999
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    sterowanie optymalne
    optimal control
    second order conditions
    semilinear elliptic PDE
    state gradient constraints
    Opis:
    The aim of this paper is to state the second order necessary and sufficient optimality conditions for distributed control problems governed by the Neumann problem associated to a semilinear elliptic partial differential equation. Bound constraints ou control are considered, as well as equality and inequality constraints of integral type on the gradient of the state.
    Źródło:
    Control and Cybernetics; 1999, 28, 3; 463-479
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations
    Autorzy:
    Osmolovskii, N. P.
    Maurer, H.
    Powiązania:
    https://bibliotekanauki.pl/articles/969736.pdf
    Data publikacji:
    2007
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    bang-bang control
    second order necessary and sufficient conditions
    critical cones
    equivalence of quadratic forms
    representation of the Hessian of the Lagrangian
    Opis:
    In Part 1 of this paper (Osmolovskii and Maurer, 2005), we have summarized the main results on the equivalence of two quadratic forms from which second order necessary and sufficient conditions can be derived for optimal bang-bang control problems. Here, in Part 2, we give detailed proofs and elaborate explicit relations between Lagrange multipliers and elements of the critical cones in both approaches. The main analysis concerns the derivation of formulas for the first and second order derivatives of trajectories with respect to variations of switching times, initial and final time and initial point. This leads to explicit representations of the second order derivatives of the Lagrangian for the induced optimization problem. Based on a suitable transformation, we obtain the elements of the Hessian of the Lagrangian in a form which involves only first order variations of the nominal trajectory. Finally, a careful regrouping of all terms allows us to find the desired equivalence of the two quadratic forms.
    Źródło:
    Control and Cybernetics; 2007, 36, 1; 5-45
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On relations among the generalized second-order directional derivatives
    Autorzy:
    Pastor, Karel
    Powiązania:
    https://bibliotekanauki.pl/articles/729322.pdf
    Data publikacji:
    2001
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    generalized second-order directional derivative
    convexity
    second-order optimality conditions
    Opis:
    In the paper, we deal with the relations among several generalized second-order directional derivatives. The results partially solve the problem which of the second-order optimality conditions is more useful.
    Źródło:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization; 2001, 21, 2; 235-247
    1509-9407
    Pojawia się w:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Revisiting the analysis of optimal control problems with several state constraints
    Autorzy:
    Bonnans, J. F.
    Hermant, A.
    Powiązania:
    https://bibliotekanauki.pl/articles/970908.pdf
    Data publikacji:
    2009
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    optimal control
    state constraints
    shooting algorithm
    second order optimality conditions
    Opis:
    This paper improves the results of and gives shorter proofs for the analysis of state constrained optimal control problems than presented by the authors in Bonnans and Hermant (2009b), concerning second order optimality conditions and the well-posedness of the shooting algorithm. The hypothesis for the second order necessary conditions is weaker, and the main results are obtained without reduction to the normal form used in that reference, and without analysis of high order regularity results for the control. In addition, we provide some numerical illustration. The essential tool is the use of the "alternative optimality system".
    Źródło:
    Control and Cybernetics; 2009, 38, 4A; 1021-1052
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Sensitivity analysis for optimal control of problems governed by semilinear parabolic equations
    Autorzy:
    Bergounioux, M.
    Merabet, N.
    Powiązania:
    https://bibliotekanauki.pl/articles/206854.pdf
    Data publikacji:
    2000
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    analiza czułości
    sterowanie optymalne
    optimal control
    second order optimality conditions
    sensitivity analysis
    Opis:
    We investigate an optimal control problem governed by a semilinear parabolic equation with perturbed initial data. We perform some sensitivity analysis: under polyhedricity assumption and second order optimality conditions we derive second order expansion of the optimal value function.
    Źródło:
    Control and Cybernetics; 2000, 29, 4; 861-886
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations
    Autorzy:
    Casas, E.
    Dhamo, V.
    Powiązania:
    https://bibliotekanauki.pl/articles/206408.pdf
    Data publikacji:
    2011
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    optimal control
    Neumann boundary control
    quasi-linear elliptic equation
    Pontryagin principle
    second order optimality conditions
    Opis:
    First- and second-order optimality conditions are established for the boundary optimal control of quasilinear elliptic equations with pointwise constraints on the control. The theory is developed for Neumann controls in polygonal domains of dimension two. For the derivation of second-order sufficient optimality conditions, which is the main goal of this paper, the regularity of the solutions to the state equation and its linearization is studied in detail. Moreover, a Pontryagin principle is proved. The main difficulty in the analysis of these problems is the nonmonotone character of the state equation.
    Źródło:
    Control and Cybernetics; 2011, 40, 2; 457-490
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Exact penalization of pointwise constraints for optimal control problems
    Autorzy:
    Casas, E.
    Powiązania:
    https://bibliotekanauki.pl/articles/970884.pdf
    Data publikacji:
    2009
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    optimal control
    semilinear elliptic equations
    pointwise state constraints
    exact penalization
    first and second order optimality conditions
    Opis:
    In this paper we consider a control problem governed by a semilinear elliptic equation with pointwise control and state constraints. We analyze the existence of an exact penalization of the state constraints. In particular, we prove that the first and second order optimality conditions imply the existence of such a penalization. Finally, we prove some extra regularity of the strict local minima of the control problem, assuming the existence of an exact penalization for them.
    Źródło:
    Control and Cybernetics; 2009, 38, 4A; 1131-1150
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second order convexity and a modified objective function method in mathematical programming
    Autorzy:
    Antczak, T.
    Powiązania:
    https://bibliotekanauki.pl/articles/969778.pdf
    Data publikacji:
    2007
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    programowanie matematyczne
    mathematical programming
    second order modified objective function optimization problem
    second order convex function
    second order optimality conditions
    Opis:
    An approach to nonlinear constrained mathematical programming problems which makes use of a second order derivative is presented. By using a second order modified objective function method, a modified optimization problem associated with a primal mathematical programming problem is constructed. This auxiliary optimization problem involves a second order approximation of an objective function constituting the primal mathematical programming problem. The equivalence between the original mathematical programming problem and its associated modified optimization problem is established under second order convexity assumption. Several practical O.R. applications show that our method is efficient. Further, an iterative algorithm based on this approach for solving the considered nonlinear mathematical programming problem is given for the case when the functions constituting the problem are second order convex. The convergence theorems for the presented algorithm are established.
    Źródło:
    Control and Cybernetics; 2007, 36, 1; 161-182
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations
    Autorzy:
    Roubicek, T.
    Troltzsch, F.
    Powiązania:
    https://bibliotekanauki.pl/articles/970527.pdf
    Data publikacji:
    2003
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    sterowanie strumieniem danych
    stabilność Lipschitza
    incompressible viscous fluids
    flow control
    first and second order optimality conditions
    Lipschitz stability
    Opis:
    An optimal control problem with quadratic cost functional for the steady-state Navier-Stokes equations with no-slip boundary condition is considered. Lipschitz stability of locally optimal controls with respect to certain perturbations of both the cost functional and the equation is proved provided a second-order sufficient optimality condition holds. For a sufficiently small Reynolds number, even global Lipschitz stability of the unique optimal control is shown.
    Źródło:
    Control and Cybernetics; 2003, 32, 3; 683-705
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    C^11 vector optimization problems and Riemann derivatives
    Autorzy:
    Ginchev, I.
    Guerraggio, A.
    Rocca, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/970359.pdf
    Data publikacji:
    2004
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    optymalizacja wektorowa
    pochodna kierunkowa typu Riemanna
    warunek optymalności dwuwskaźnikowy
    vector optimization
    Riemann-type directional derivatives
    second-order optimality conditions
    Opis:
    In this paper we introduce a generalized second-order Riemann-type derivative for C^1'1 vector functions and use it to establish necessary and sufficient optimality conditions for vector optimization problems. We show that, these conditions are stronger than those obtained by means of the second-order subdinerential in Clarke sense considered in Guerraggio, Luc (2001) and also to some extent than those obtained in Guerraggio, Luc, Minh (2001).
    Źródło:
    Control and Cybernetics; 2004, 33, 2; 259-273
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Network optimality conditions
    Autorzy:
    Osmolovskii, Nikolai P.
    Qian, Meizhi
    Sokołowski, Jan
    Powiązania:
    https://bibliotekanauki.pl/articles/31343935.pdf
    Data publikacji:
    2023
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    network
    optimal control problem
    weak local minimum
    Pontryagin’s maximum principle
    critical cone
    quadratic form
    second order optimality conditions
    Riccati equation
    Opis:
    Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the local Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables xi varying within their intervals [0, li], associated with the corresponding beams at network edges. The lengths li of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.
    Źródło:
    Control and Cybernetics; 2023, 52, 2; 129-180
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint
    Autorzy:
    Osmolovskii, Nikolai P.
    Powiązania:
    https://bibliotekanauki.pl/articles/2183496.pdf
    Data publikacji:
    2022
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    critical cone
    quadratic form
    first order tangent
    second order tangent
    second order optimality conditions
    weak local minimum
    inequality control constraint
    Pontryagin’s maximum principle
    Opis:
    This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0, given by a C2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set U, defined by the inequality G(u) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient Hu of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.
    Źródło:
    Control and Cybernetics; 2022, 51, 2; 151--169
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Extremals of the time optimal control problem for a material point moving along a straight line in the presence of friction and limitation on the velocity
    Autorzy:
    Osmolovskii, N. P.
    Figura, A.
    Kośka, M.
    Wójtowicz, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/206453.pdf
    Data publikacji:
    2017
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    material point
    resistance forces
    acceleration
    deceleration
    Pontryagin’s maximum principle
    second order optimality conditions
    optimal control
    adjoint variable
    state constraint
    Stieltjes measure
    singular arc
    boundary arc
    Opis:
    This paper provides an analysis of the time optimal control problem for a material point moving along a straight line in the presence of strength of resistance to movement (friction) and subject to constraint on the velocity. The point is controlled by a limited traction or braking force. The analysis of the problem is based on the maximum principle for state constraints in the Dubovitskii-Milyutin form, see Dubovitskii and Milyutin (1965), and the necessary second-order optimality condition for bang-bang controls, see Milyutin and Osmolovskii (1998).
    Źródło:
    Control and Cybernetics; 2017, 46, 4; 305-324
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second-order necessary conditions for discrete inclusions with end point constraints
    Autorzy:
    Cernea, Aurelian
    Powiązania:
    https://bibliotekanauki.pl/articles/729636.pdf
    Data publikacji:
    2005
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    tangent cone
    discrete inclusion
    necessary optimality conditions
    Opis:
    We study an optimization problem given by a discrete inclusion with end point constraints. An approach concerning second-order optimality conditions is proposed.
    Źródło:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization; 2005, 25, 1; 47-58
    1509-9407
    Pojawia się w:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second-order conditions for boundary control problems of the Burgers equation
    Autorzy:
    Volkwein, S.
    Powiązania:
    https://bibliotekanauki.pl/articles/206716.pdf
    Data publikacji:
    2001
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    równanie Burgersa
    stan optymalności
    sterowanie optymalne
    wielościenność
    Burgers' eguation
    optimal control
    optimality conditions
    polyhedricity
    Opis:
    In this article control constrained optimal control problems for the Burgers equation are considered. First- and second-order optimality conditions are presented. Utilizing polyhedricity of the feasible set and the theory of Legendre-forms a second-order sufficient optimality condition is given that is very close to the second-order necessary optimality condition. For the numerical realization a prima-dual actrive set strategy is used.
    Źródło:
    Control and Cybernetics; 2001, 30, 3; 249-278
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Boundary integral representations of second derivatives in shape optimization
    Autorzy:
    Eppler, Karsten
    Powiązania:
    https://bibliotekanauki.pl/articles/729245.pdf
    Data publikacji:
    2000
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    optimal shape design
    fundamental solution
    boundary integral equation
    second-order derivatives
    optimality conditions
    Opis:
    For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sufficient optimality conditions.
    Źródło:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization; 2000, 20, 1; 63-78
    1509-9407
    Pojawia się w:
    Discussiones Mathematicae, Differential Inclusions, Control and Optimization
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Sufficient optimality conditions for a bang-singular extremal in the minimum time problem
    Autorzy:
    Poggiolini, L.
    Stefani, G.
    Powiązania:
    https://bibliotekanauki.pl/articles/969613.pdf
    Data publikacji:
    2008
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    minimum time
    second order sufficient conditions
    bang-singular arc
    Hamiltonian formalism
    Opis:
    The paper gives second order sufficient conditions for the strong local optimality of a bang-singular extremal in a minimum time problem. The conditions are given in terms of regularity assumptions on the extremal and of the coercivity of the extended second variation associated to the minimum time problem with fixed end-points on the singular arc. The conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where necessary conditions have mild inequalities.
    Źródło:
    Control and Cybernetics; 2008, 37, 2; 469-490
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Second-order sufficient conditions for an extremum in optimal control
    Autorzy:
    Osmolovskij, N.
    Powiązania:
    https://bibliotekanauki.pl/articles/206727.pdf
    Data publikacji:
    2002
    Wydawca:
    Polska Akademia Nauk. Instytut Badań Systemowych PAN
    Tematy:
    ekstremala łamana
    forma kwadratowa
    minimum silne
    minimum słabe
    stożek krytyczny
    warunek dostatecznej optymalności
    zasada maksimum
    accessory problem
    broken extremal
    control discontinuities
    critical cone
    maximum principle
    quadratic form
    strong minimum
    suffcient optimality condition
    weak minimum
    Opis:
    Suffcient quadratic optimality conditions for a weak and a strong minimum are stated in an optimal control problem on a fixed time interval with mixed state-control constraints, under the assumption that the gradients of all active mixed constraints with respect to control are linearly independent. The conditions are stated for the cases of both continuous and discontinuous controls and guarantee in each case a lower bound of the cost function increase at t1e reference point. They are formulated in terms of an accessory problem with quadratic form, which must be positive-definite on the so-called critical cone. In the case of discontinuous control the quadratic form has some new terms related to the control discontinuity.
    Źródło:
    Control and Cybernetics; 2002, 31, 3; 803-831
    0324-8569
    Pojawia się w:
    Control and Cybernetics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-22 z 22

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