Kelley condition and structure of Lagrange manifold in a neighborhood of a first-order singular extremal

The paper considers optimal control problems linearly depending on the scalar control parameter in which there exist first-order singular extremals. The author proves a theorem on the structure of a generic Lagrange manifold (field of extremals) in a neighborhood of first-order singular extremals. As a consequence of this theorem, the author proves the optimality of singular extremals and nonsingular extremals in problems with fixed endpoints on small intervals of time. As an illustration, the paper presents constructions of Lagrange manifolds for the general linear-quadratic control problem with completely integrable linear system of differential constraints and for a certain problem of mathematical economics, a two-factor economic growth model with production function of the Cobb-Douglas type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Generic structure of the Lagrangian manifold in chattering problems

This paper studies the structure of the singularity of Lagrangian manifolds in a neighborhood of the surface of singular extremals of second order in optimal control problems. For the Fuller classical problem, the structure of the Lagrangian manifold is explicitly constructed: it is shown that it has a singularity of conic type at the origin of coordinates. In the general case, it is proved that the Lagrangian manifold is a locally trivial fiber bundle over the surface of singular extremals with each fiber having a singularity of a similar conic type at the point of exit of the singular extremals.     

Domination by positive disjointly strictly singular operators

We prove that each positive operator from a Banach lattice to a Banach lattice with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on is order continuous. We prove as well that if is dominated by a disjointly strictly singular operator, then is disjointly strictly singular.

     

L p -weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups

In this paper, we consider a large class of critical problems with singular potentials on Carnot groups. In particular, we prove sharp regularity results in Marcinkiewicz spaces and the exact rate of decay of the solutions. Our general results apply, for instance, to the equation satisfied by the extremals of some relevant Hardy–Sobolev type inequalities on Stratified groups.     

A global bifurcation phenomenon for second order singular boundary value problems

We establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem. As an application, we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems.     

Optimal synthesis in an infinite-dimensional space

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space l ₂, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space l ₂ forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.     

Extremals with accumulation of switchings in an infinite-dimensional space

This paper studies a class of optimal control problems and the Hamiltonian systems generated by them in the space l ₂. The authors prove the existence of extremals with countably many switchings on a finite interval of time that compose a bundle with piecewise fibers over the base of singular extremals of condimension 4. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 58, Optimal Control, 2008.     

Junction times in singular optimal control

Finding junction times on partially singular optimal control trajectories is in general difficult analytically and especially so when the optimal control is explicitly dependent on both the state and adjoint variables. This paper presents a new approach for linear problems whereby variations in the final state, measured from a reference optimal solution, can be used to find corresponding junction times. The approach is shown to be effective when applied to a problem which has an optimal solution involving a single bang/singular junction for certain choices of end state.     

Jacobi's condition for singular extremals: An extended notion of conjugate points

For the simple fixed endpoint problem in the calculus of variations, Jacobi's condition (“there are no conjugate points in the interior of the underlying time interval”) is necessary for optimality if the trajectory under consideration is nonsingular. In this paper, we extend the notion of conjugate points so that the above condition (in terms of this new notion) is necessary also for singular extremals. This is achieved by showing that, without any additional assumption on the trajectory, the nonnegativity of the second variation on the space of admissible variations is equivalent to the nonexistence of these “extended conjugate points”.     

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus iff and . The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

     

Twin solutions to singular semipositone problems

In this paper, by a specially constructed cone and the fixed point index theory, we investigate the existence of multiple positive solutions for the following singular semipositone problem:

     

Nontrivial solutions of singular sublinear Sturm-Liouville problems

The singular sublinear Sturm-Liouville problems

     

Singular Normal Extremals and Conjugate Points for Bolza Functionals

In this paper, we study the problem of the optimal control for Bolza functionals by investigating extremals containing singular arcs. We use the Moore–Penrose generalized inverse, which allows one to determine normality criteria and sufficient conditions for the nonexistence of conjugate points.     

Lagrange multipliers for regular and singular problems in the calculus of variations

A Lagrange multiplier rule is presented for a variational problem of Bolza type under hypotheses that allow certain components of the coefficient matrices involved in the functional being minimized to fail to be integrable near an endpoint of the interval on which the relevant functions are defined. The problem is also addressed when all coefficients are of classL ², but not necessarily bounded. Applications are made to ascertain properties of functions providing equality to certain singular and regular integral inequalities appearing in the literature.     

Nontrivial solutions of singular superlinear Sturm-Liouville problems

The singular superlinear Sturm-Liouville problems

     

Linearizability of Poisson structures around singular symplectic leaves

The linearization problem for a Poisson structure near a singular symplectic leaf of nonzero dimension is studied. We obtain the following generalization of the Conn linearization theorem: if the transverse Lie algebra of the leaf is semisimple and compact, then the Poisson structure is linearizable, provided that certain cohomological obstructions vanish.     

The first focus quantity of a complex monodromic singular point

We consider the problem of distinguishing the center and focus in the case of a strongly degenerate singular point. A sufficient condition is explicitly stated for the existence of a focus for singular points of vector fields on the plane belonging to a certain class.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 106–122, 1988.