Vol 304, No 1 (2019)

We briefly describe the history of the discovery of the Pontryagin maximum principle in optimal control theory in the mid-1950s by Academician L. S. Pontryagin in collaboration with his students—V. G. Boltyanskii and the author of the present paper—who were then young scientists at the Department of Differential Equations headed by Pontryagin at the Steklov Mathematical Institute.

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the classical Euler identity Π_n=1^∞(1 − x²/(ρn)²) = sin x/x. The general case may serve as a rich source of new nice identities.

Existence and uniqueness theorems are obtained for a fixed point of a mapping from a complete metric space to itself. These theorems generalize the theorems of L. V. Kantorovich for smooth mappings of Banach spaces. The results are extended to the coincidence points of both ordinary and set-valued mappings acting in metric spaces.

We consider an infinite-horizon optimal control problem with an integral objective functional containing a discount factor in the integrand. A specific feature of the problem is the assumption that the integrand may be unbounded. The main result of the paper is an estimate of the approximation accuracy in a backward procedure for solving the Hamilton-Jacobi equation corresponding to the optimal control problem.

For a tuple A = (A₁,A₂,…, A_n) of elements in a unital Banach algebra B, the associated multiparameter pencil is A(z) = z₁A₁ + z₂A₂ + … + z_nA_n. The projective spectrum P(A) is the collection of z ∈ ℂⁿ such that A(z) is not invertible. Using the fundamental form Ω_A = −ω_A^* ∧ ω_A, where ω_A(z) = A⁻¹(z) dA(z) is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set P^c(A) = ℂⁿ \ P(A). This paper examines that metric in the case of the infinite dihedral group, D_∞ = <a, t | a² = t² = 1>, with respect to the left regular representation λ. For the non-homogeneous pencil R(z) = I + z₁λ(a) + z₂λ(t), we explicitly compute the metric on P^c(R) and show that the completion of P^c(R) under the metric is ℂ² \ {(±1, 0), (0, ±1)}, which rediscovers the classical spectra σ(λ(a)) = σ(λ(t)) = {± 1}. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).

This article is a continuation of the author’s previous paper of 2015. It deals with electromagnetic phenomena in a model of a flat two-dimensional ring with a given law of differential rotation. The reasons of high electromagnetic activity in Saturn's ring are discussed. In parallel with the discussion, it is shown that similar reasons can clarify other processes such as the charging of thunderclouds. New aspects in the theory of magnetic fields of planets are also proposed.

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.

The problem of tracking a solution of a nonlinear system of ordinary differential equations is considered in the case of inaccurate measurement of some of the phase coordinates. A noise-immune solution algorithm for this system is proposed that is based on a combination of constructs from dynamic inversion and guaranteed control theories. The algorithm consists of two blocks: a block of dynamical reconstruction of unmeasured coordinates and a feedback control block.

We develop Pontryagin’s direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler—Lagrange differential inclusion. The results are proved under maximally weak conditions, and very strong statements compared with the known ones are obtained; moreover, admissible velocity sets may be unbounded and nonconvex under a general hypothesis that the right-hand side of the differential inclusion is pseudo-Lipschitz. In the statements, we refine conditions on the Euler—Lagrange differential inclusion, in which neither the Clarke normal cone nor the limiting normal cone is used, as is common in the works of other authors. We also give an example demonstrating the efficiency of the results obtained.

We present some modifications of the definitions of a u-stable bridge and of an approximating system of sets in a game problem of approach at a fixed instant of time. These modifications are aimed at developing and substantiating algorithms for approximate construction of solutions in approach game problems.

We consider a differential game in which one of the players tries to keep a trajectory within a given set of vector functions on a finite time interval; the goal of the second player is opposite. To construct the set of successful solvability in this problem, which is defined by the functional target set, we apply the programmed iteration method. The essence of the method lies in a universal game problem of programmed control that depends on parameters characterizing the constraints on the initial fragments of trajectories. As admissible control procedures, we use multivalued quasistrategies (regarding a conflict-controlled system, it is assumed that the conditions of generalized uniqueness and uniform boundedness of programmed motions are satisfied).