"1"^^ . "2007-05-01+02:00"^^ . . . "2012-07-12+02:00"^^ . "2011-12-31+01:00"^^ . . "optimization problems" . "optimization problems; time scale symplectic systems; optimality conditions; quadratic functionals"@en . . . . "C\u00EDlem projektu je z\u00E1kl. matematick\u00FD v\u00FDzkum v oblasti podm\u00EDnek optimality ve spojit\u00FDch, diskr\u00E9tn\u00EDch a dal\u0161\u00EDch obecn\u011Bj\u0161\u00EDch optimaliza\u010Dn\u00EDch probl\u00E9mech. Proto budou studov\u00E1ny podm\u00EDnky optimality druh\u00E9ho \u0159\u00E1du, kter\u00E9 souvis\u00ED s definitnost\u00ED(tj. s nez\u00E1pornost\u00ED a koercivitou) kvadratick\u00FDch funkcion\u00E1l\u016F druh\u00E9 variace, kter\u00E9 jsou nutn\u00FDmi(nez\u00E1pornost) a posta\u010Duj\u00EDc\u00EDmi (koercivita, pozitivita) podm\u00EDnkami pro optimalitu v p\u016Fvodn\u00EDm optimaliza\u010Dn\u00EDm probl\u00E9mu. Podm\u00EDnky optimality pomoc\u00ED kvadratick\u00FDch funkcion\u00E1l\u016F druh\u00E9 variace lze formulovat ekvivalentn\u00EDm zp\u016Fsobem pomoc\u00ED konjugovan\u00FDch a sdru\u017Een\u00FDch bod\u016F \u010Di vlastnostmi \u0159e\u0161en\u00ED p\u0159\u00EDslu\u0161n\u00E9 Jacobiho nebo Riccatiho rovnice, c\u00EDlem je tedy odvozen\u00ED nov\u00FDch a zobecn\u011Bn\u00ED st\u00E1vaj\u00EDc\u00EDch zn\u00E1m\u00FDch podm\u00EDnek, kter\u00E9 charakterizuj\u00ED pozitivitu a nez\u00E1pornost takov\u00FDch kvadratick\u00FDch funkcion\u00E1l\u016F. Dal\u0161\u00ED je zam\u011B\u0159en\u00ED na zobecn\u011Bn\u00ED v\u00FDsledk\u016F na symplektick\u00E9 syst\u00E9my na %22time scales%22, jako nejobecn\u011Bj\u0161\u00EDch objekt\u016F pro n\u011B\u017E %22rozumn\u011B%22 funguje kvalitativn\u00ED teorie diferenci\u00E1ln\u00EDch, diferen\u010Dn\u00EDch \u010Di dynamick\u00FDch rovnic." . . "20"^^ . "In this project, first and second order optimality conditions for calculus of variations and optimal control problems on time scales were studied.These conditions are phrased in terms of Hamiltonian and symplectic systems and their qualitative properties. We derived new results about these systems and the optimality conditions themselves. This project contributed to 19 scientific publications."@en . " time scale symplectic systems" . "20"^^ . "http://www.isvav.cz/projectDetail.do?rowId=ME 891"^^ . "ME 891" . . "2011-03-14+01:00"^^ . " optimality conditions" . . . "Podm\u00EDnky optimality druh\u00E9ho \u0159\u00E1du pro optimaliza\u010Dn\u00ED probl\u00E9my" . . "The subject of the study is mathematical research in the area of optimality conditions in continuous, discrete, and other more general optimization problems to find the optimal conditions of the 2nd order which are connected to the definiteness(nonnegativity and coercivity)of the quadratic functional of the 2nd variation. These conditions are necessary(nonnegativity) and sufficient (coercivity,positivity) for the optimality in the original optimization problem. Above mentioned optimality conditions can be formulated in an equivalent way via conjugate and coupled points or properties of solutions of the corresponding Jacobi and Riccati equations. The aim is so deriving new and generalizing conditions to characterize the positivity and nonnegativity of such quadratic functionals. Moreover, there is focus on generalizing the results to time scale symplectic systems, which are the most general objects, for which the qualitative theory of differential, difference, and/or dynamic equations reasonably works."@en . . . "V projektu byly studov\u00E1ny podm\u00EDnky optimality prvn\u00EDho a druh\u00E9ho \u0159\u00E1du pro \u00FAlohy varia\u010Dn\u00EDho po\u010Dtu a optim\u00E1ln\u00EDho \u0159\u00EDzen\u00ED na \u010Dasov\u00FDch \u0161k\u00E1l\u00E1ch. Prost\u0159edkem jsou hamiltonovsk\u00E9 a symplektick\u00E9 syst\u00E9my a jejich kvalitativn\u00ED vlastnosti. V projektu se poda\u0159ilo odvodit nejen nov\u00E9 poznatky o t\u011Bchto syst\u00E9mech, ale i o podm\u00EDnk\u00E1ch optimality jako takov\u00FDch. V\u00FDsledkem projektu je 19 kvalitn\u00EDch v\u011Bdeck\u00FDch publikac\u00ED."@cs . . . . "Second order optimality conditions for optimization problems"@en . . "0"^^ . . "0"^^ .