Differential Equations最新文献

{"title":"On Regularization of Classical Optimality Conditions in Convex Optimization Problems for Volterra-Type Systems with Operator Constraints","authors":"V. I. Sumin, M. I. Sumin","doi":"10.1134/s0012266124020071","DOIUrl":"https://doi.org/10.1134/s0012266124020071","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the regularization of classical optimality conditions—the Lagrange\u0000principle and the Pontryagin maximum principle—in a convex optimal control problem\u0000with an operator equality constraint and functional inequality constraints. The controlled system\u0000is specified by a linear functional–operator equation of the second kind of general form in the\u0000space <span>(L^m_2 )</span>, and the main operator on the right-hand side of\u0000the equation is assumed to be quasinilpotent. The objective functional of the problem is only\u0000convex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is\u0000based on the dual regularization method. In this case, two regularization parameters are used, one\u0000of which is “responsible” for the regularization of the dual problem, and the other is contained in\u0000the strongly convex regularizing Tikhonov addition to the objective functional of the original\u0000problem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function.\u0000The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the\u0000stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized\u0000classical optimality conditions\u0000</p><ol>\u0000<li>\u0000<span>1.</span>\u0000<p>Are formulated as existence theorems for minimizing approximate solutions in the original\u0000problem with a simultaneous constructive representation of these solutions. </p>\u0000</li>\u0000<li>\u0000<span>2.</span>\u0000<p>Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.\u0000</p>\u0000</li>\u0000<li>\u0000<span>3.</span>\u0000<p>“Overcome” the properties of the ill-posedness of the classical optimality conditions and\u0000provide regularizing algorithms for solving optimization problems. </p>\u0000</li>\u0000</ol><p>Based on the perturbation method, an important property of the regularized\u0000classical optimality conditions obtained in the work is discussed in sufficient detail; namely, “in the\u0000limit” they lead to their classical counterparts. As an application of the general results obtained in\u0000the paper, a specific example of an optimal control problem associated with an integro-differential\u0000equation of the transport equation type is considered, a special case of which is a certain inverse\u0000final observation problem.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"52 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Internal Transition Layer Structure in the Reaction–Diffusion Problem for the Case of a Balanced Reaction with a Weak Discontinuity","authors":"E. I. Nikulin, V. T. Volkov, D. A. Karmanov","doi":"10.1134/s0012266124010063","DOIUrl":"https://doi.org/10.1134/s0012266124010063","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a singularly perturbed reaction–diffusion equation, we study the structure of the\u0000internal transition layer in the case of a balanced reaction with a weak discontinuity. The\u0000existence of solutions with an internal transition layer (contrast structures) is proved, the question\u0000of their stability is investigated, and asymptotic approximations to solutions of this type are\u0000obtained. It is shown that in the case of reaction balance, the presence of even a weak\u0000(asymptotically small) reaction discontinuity can lead to the formation of contrast structures of\u0000finite size, both stable and unstable.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"28 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reflecting Function and a Generalization of the Notion of First Integral","authors":"V. I. Mironenko, V. V. Mironenko","doi":"10.1134/s0012266124010026","DOIUrl":"https://doi.org/10.1134/s0012266124010026","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The relationships between the notion of generalized integral and the notions of reflecting\u0000function and Poincaré map (period map) for periodic differential systems are traced.\u0000The notion of generalized first integral is used to study questions of the existence and stability of\u0000periodic solutions of periodic differential systems and analyze the center–focus problem.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems","authors":"M. G. Yumagulov, L. S. Ibragimova","doi":"10.1134/s0012266124010038","DOIUrl":"https://doi.org/10.1134/s0012266124010038","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New approaches are proposed in the problem of constructing equivalent scalar differential\u0000equations for multidimensional nonlinear systems of control theory, as well as in the problem of\u0000constructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differential\u0000equations containing derivatives of only even orders). The conditions for the solvability of the\u0000corresponding problems are studied, and new formulas for the transition to equivalent equations\u0000and systems are proposed. For the Lurie equations, the proposed approaches are based on the\u0000transition from the linear part to the normal forms of the corresponding Hamiltonian systems with\u0000a subsequent transformation of the resulting system. Calculation formulas and algorithms are\u0000obtained, and their efficiency is illustrated by examples.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"107 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Applying Differential-Geometric Control Theory Methods in the Theory of Partial Differential Equations. III","authors":"V. I. Elkin","doi":"10.1134/s0012266124010051","DOIUrl":"https://doi.org/10.1134/s0012266124010051","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the symmetries of partial differential equations based on the use of\u0000differential-geometric and algebraic methods of the theory of dynamical control systems.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"5 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance","authors":"K. A. Shchelchkov","doi":"10.1134/s0012266124010105","DOIUrl":"https://doi.org/10.1134/s0012266124010105","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the problem of stabilization to zero under disturbance in terms of\u0000a differential pursuit game. The dynamics is described by a nonlinear autonomous system of\u0000differential equations. The set of control values of the pursuer is finite, and that of the evader\u0000(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring the\u0000trajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.\u0000To construct the control, the pursuer knows only the state coordinates at some discrete times, and\u0000the choice of the disturbance’s control is unknown. In the paper, we obtain conditions for the\u0000existence of a neighborhood of zero from each point of which a capture occurs in the indicated\u0000sense. A winning control is constructed constructively and has an additional property specified in\u0000a theorem. In addition, an estimate of the capture time sharp in some sense is produced.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"115 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectrum Assignment for a System of Neutral Type","authors":"A. V. Metel’skii","doi":"10.1134/s0012266124010099","DOIUrl":"https://doi.org/10.1134/s0012266124010099","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a linear autonomous system of neutral type with commensurable delays, an algorithm\u0000is given for solving the modal controllability problem (in particular, the finite spectrum\u0000assignment problem), which provides a closed-loop system with a given characteristic\u0000quasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterion\u0000for exponential stabilization of the system under study is constructively justified. When the\u0000criterion is met, the closed-loop system can be made exponentially stable according to the\u0000proposed spectral reduction algorithm. The obtained statements and spectrum assignment\u0000algorithms are illustrated with examples.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Method of Lyapunov Functionals and the Boundedness of Solutions and Their First and Second Derivatives for a Third-Order Linear Equation of the Volterra Type on the Half-Line","authors":"S. Iskandarov, A. T. Khalilov","doi":"10.1134/s0012266124010087","DOIUrl":"https://doi.org/10.1134/s0012266124010087","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Sufficient conditions are established for the boundedness of all solutions and their first two\u0000derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line.\u0000To this end, using a method proposed by the first author in 2006, first, we reduce the equation\u0000under consideration to an equivalent system consisting of one first-order differential equation and\u0000one second-order Volterra integro-differential equation. Then a new generalized Lyapunov\u0000functional is proposed for this system, the nonnegativity of this functional on solutions of this\u0000system is proved, and an upper bound is given for the derivative of this functional via the original\u0000functional. The resulting estimate is an integro-differential inequality whose solution gives an\u0000estimate of the functional.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"39 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inverse Problem of Determining Two Coefficients of Lower-Order Terms in a Mixed Parabolic-Hyperbolic Equation","authors":"D. K. Durdiev","doi":"10.1134/s001226612401004x","DOIUrl":"https://doi.org/10.1134/s001226612401004x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Direct and inverse problems for a model equation of mixed parabolic-hyperbolic type are\u0000studied. In the direct problem, we consider a Tricomi-type problem for this equation with a\u0000noncharacteristic line of type change. The unknowns of the inverse problem are the variable\u0000coefficients of the lower-order terms in the equation. To determine these coefficients, an integral\u0000overdetermination condition is specified relative to the solution defined in the parabolic part of the\u0000domain, and in the hyperbolic part, conditions are specified on the characteristics: on one\u0000characteristic it is the value of the normal derivative and on the other, the value of the function\u0000itself. Theorems for the unique solvability of the posed problems in the sense of classical solution\u0000are proved.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"64 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solutions of Analogs of Time-Dependent Schrödinger Equations Corresponding to a Pair of $$H^{2+2+1}$$ Hamiltonian Systems in the Hierarchy of Degenerations of an Isomonodromic Garnier System","authors":"V. A. Pavlenko","doi":"10.1134/s0012266124010075","DOIUrl":"https://doi.org/10.1134/s0012266124010075","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> This paper continues a series of papers in which simultaneous <span>(2times 2 )</span> matrix solutions of two scalar evolution equations,\u0000which are analogs of time-dependent Schrödinger equations, were constructed. In the\u0000constructions in the present paper, these equations correspond to the Hamiltonian system\u0000<span>(H^{2+2+1} )</span>—one of the representatives of the hierarchy\u0000of degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described by\u0000H. Kimura in 1986. In terms of solutions of linear systems of differential equations in the method\u0000of isomonodromic deformations, the consistency condition for which is the Hamiltonian equations\u0000of the <span>(H^{2+2+1} )</span> system, the constructed simultaneous matrix\u0000solutions of analogs of time-dependent Schrödinger equations are written out explicitly in\u0000this paper.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"39 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}