Pontryagin Function Research Articles

USE OF ANALYTICAL MODEL FOR SYNTHESIS OF ALGORITHMS FOR CONTROL OF TRANSPORT CONVEYOR PARAMETERS

This study presents a methodology for synthesizing optimal control algorithms for the flow parameters of a conveyor-type transport system with a variable transport delay. A multi-section transport conveyor is a complex dynamic system with a variable transport delay. The transport conveyor is an important element of the production system, used to synchronize technological operations and move material. The Analytical PiKh-model of the conveyor section was used as a model for designing a control system for flow parameters. The characteristic dimensionless parameters of the conveyor section are introduced and the similarity criteria for the conveyor sections are determined. The model of a conveyor section in a dimensionless form is used to develop a methodology for synthesizing algorithms for optimal control of the flow parameters of a transport conveyor section. The dependencies between the value of the input and output material flow of the section are determined, taking into account the initial distribution of the material along the conveyor section, variable transport delay, restrictions on the specific density of the material, and restrictions on the speed of the belt. The dependencies between the value of the input and output material flow for the case of a constant transport delay are analyzed. A technique for synthesizing algorithms for optimal belt speed control based on the PiKh-model of a conveyor section is presented. As a simplification, a two-stage belt speed control is considered. Particular attention is paid to the methodology for synthesizing optimal control algorithms based on the energy management methodology (TOU-Tariffs). The criteria of control quality are introduced and problems of optimal control of flow parameters of the transport system are formulated. Taking into account differential connections and restrictions on phase variables and admissible controls, which are typical for the conveyor section, the Pontryagin function and the adjoint system of equations are written. As examples demonstrating the design of optimal control, algorithms for optimal control of the flow parameters of the transport system are synthesized and analysis of optimal controls is performed.

Позиционный принцип минимума: вариационное усиление понятий экстремальности в оптимальном управлении

Existing maximum principles of Pontryagin’s type and related optimality conditions, such as, e.g., the ones derived by F. Clarke, B. Kaskosz and S. Lojasiewicz Jr., and H.J. Sussmann, can be strengthened up to global necessary optimality conditions in the form of so-called feedback minimum principle. This is possible for both classical and non-smooth optimal control problems without terminal constraints. The formulation of the feedback minimum principle (or related extremality conditions) remains within basic constructions of the mentioned maximum principles (the Hamiltonian or Pontryagin function, the adjoint differential equation or inclusion, and its solutions –– co-trajectories). At the same time, the actual maximum condition –– maximization of the Hamiltonian –– takes a variational form: any optimal trajectory of the addressed problem should be optimal for a specific “accessory” problem of dynamic optimization. The latter is stated over all tubes of Krasovskii-Subbotin constructive motions generated by feedback strategies, which are extremal with respect to a certain supersolution of the Hamilton-Jacobi equation. Such a supersolution can be represented explicitely in terms of the co-trajectory of a reference control process and the terminal cost function. In a general version, the feedback minimum principle operates with generalized solutions of the proximal Hamilton-Jacobi inequality for weakly decreasing (𝑢-stable) functions

On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

Necessary Optimality Conditions for Optimal Control Problems in the Presence of Degeneration

We study optimal control problems in which the condition of maximum of the Hamilton–Pontryagin function degenerates for the optimal pair at some point. New additional necessary conditions are obtained for such problems.

О разрешимости задачи нелинейной оптимизации колебательных процессов при появлении особых управлений

The article deals with the cases when special controls appear in nonlinear optimization of oscillatory processes, when the function of external influence depends non-linearly on the control parameter. Quality-managed process is to minimize the integral of the functional, that is, in the final moment of time square deviation of a controlled process from a given desired state was minimal. The study was conducted using a generalized solution of the boundary value problem, which more or less adequately describes the actual process. According to the well-known method of optimal control theory, the increment of the functional is calculated and the Pontryagin function is written out, which is studied for the maximum in the range of acceptable values of the control function. The optimal control conditions are written out in the form of equality and differential inequality, which must be fulfilled simultaneously. The case when the maximum principle degenerates is studied. It is established that the desired control satisfies an infinite-dimensional system of linear integral Fredholm equations of the first kind, the solvability of which is studied by operator methods. It is proved that the operator equation has infinitely many solutions and an algorithm for their construction is developed. Then the functional is minimized on the set of solutions of the operator equation. This problem has at least one solution that is the desired special optimal control.

Inverse Problems in Economic Measurements

The problem of economic measurements is discussed. The system of economic indices must reflect the economic relations and mechanisms existing in society. An achievement of the XX century is the development of a system of national accounts and the gross domestic product index. However, the gross domestic product index, which is related to the Hamilton–Pontryagin function in extensive economic growth models, turns out to be inadequate under the conditions of structural changes. New problems of integral geometry related to production models that take into account the substitution of production factors are considered.

An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals

We prove a theorem on necessary conditions of Pontryagin's maximum principle type for an optimum of functionals given by multiple integrals. In contrast to the case of one- dimensional integrals, the maximum of the Pontryagin function is taken only over matrices of rank , not over all matrices. We give some examples.

Optimal control of biomass from the selection apparatus

The variant of solving the problem of optimal control of biomass selection from the apparatus using the Pontryagin maximum principle is presented in the article. The maximum productivity of the device as an optimality criterion was chosen. On a finite-dimensional space the Pontryagin function, the Hamiltonian function, and the end function were defined. To describe the dynamics of the development of the yeast population, the Ferhulst-Gause equation was used. It is proved that optimal control (in the chosen criterion) is possible with the found ratios of the coefficient of natural growth e and the coefficient β taking into account the intraspecies competition of microorganisms. To determine the optimal concentration of microorganisms and the biomass selection value, it is necessary to determine the numerical values of the coefficients of the model e and β during the cultivation process. The method for determining the model coefficients and the optimal biomass selection from the apparatus was presented. The algorithm and the system for optimal control of the selection of biomass from the apparatus were developed. They differ in that here the found theoretical relationships of the named coefficients are used. To study the optimal control system, a computer experiment was carried out using experimental data on the hourly accumulation of microorganisms taken at the Voronezh yeast plant. The data were obtained for a 13 hour culture process with selection of the culture medium after 6 hours. The selection is conducted in such a way that the accumulation (yeast concentration) of X during sampling remains unchanged with a constant culture medium selection. That is, the volume of biomass equal to the volume of the medium supplied with the feeding is taken. The graphs of the hourly change in yeast concentration and the dynamics of selection were obtained. The results analysis shows that the use of such a system in the process of controlling yeast production makes it possible to shorten the cultivation time by 1 hour.

Sufficient optimality conditions for extremal controls based on functional increment formulas

Optimal control problem without phase and terminal constraints is considered. Conceptions of strongly extremal controls are introduced on the basis of nonstandard functional increment formulas. Such controls are optimal in linear and quadratic problems. In general case optimality property is guaranteed by concavity condition of the Pontryagin function with respect to phase variables.

Synchronization of traveling waves in a dispersive system of weakly coupled equations

The system of weakly coupled differential equations describing traveling waves in dispersive media is considered. The Lyapunov — Schmidt construction is used to study the branching of cnoidal-type periodic solutions. The analysis of bifurcation equations uses the group symmetry and cosymmetry of original equations. Sufficient condition for existence of the phase-shifted modes of cnoidal waves is formulated in terms of the Pontryagin's function determined by the nonlinear perturbation terms

Principal Poincaré Pontryagin function associated to some families of Morse real polynomials

It is known that the principal Poincaré Pontryagin function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is also an Abelian integral in non-generic cases.In non-generic cases it is an iterated integral. Uribe (2006 J. Dyn. Control. Syst. 12 109–34, 2009 J. Diff. Eqns 246 1313–41) gives in a special case a precise description of the principal Poincaré Pontryagin function, an iterated integral of length at most 2, involving logarithmic functions with only 1 ramification at a point at infinity. We extend this result to some non-isomonodromic families of real Morse polynomials.

Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of [formula omitted] straight lines

In this paper, we study small polynomial perturbations of a Hamiltonian vector field with Hamiltonian F formed by a product of ( d + 1 ) real linear functions in two variables. We assume that the corresponding lines are in a general position in R 2 . That is, the lines are distinct, non-parallel, no three of them have a common point and all critical values not corresponding to intersections of lines are distinct. We prove in this paper that the principal Poincaré–Pontryagin function M k ( t ) , associated to such a perturbation and to any family of ovals surrounding a singular point of center type, belongs to the C [ t , 1 / t ] -module generated by Abelian integrals and some integrals I i , j ∗ ( t ) , with 1 ⩽ i < j ⩽ d defined in the paper. Moreover, I i , j ∗ ( t ) are not Abelian integrals. They are iterated integrals of length two.

Principal Poincaré–Pontryagin Function of Polynomial Perturbations of the Hamiltonian Triangle

In this paper, we consider a small polynomial perturbation of the Hamiltonian vector field with the Hamiltonian F ( x, y ) = x [ y 2 ( x 3) 2 ] having a center bounded by a triangle. The main resul...

The method of multiple maxima and optimality conditions for singular extremals

We consider optimal control problems that can have singular or sliding modes as solutions or parts of solutions [1; 2, p. 26]; in these modes, the Pontryagin function is independent of some control variables or has a nonunique maximum on the set of control variables. For such problems with specific degeneration properties [3, p. 47], the classical optimality conditions (if they apply at all) are noneffective: the Lagrange–Pontryagin necessary conditions are satisfied and can produce infinitely many spurious solutions, while the sufficient conditions fail to hold. One needs effective conditions, which are lacking in the theory yet. The aim of the present paper is to fill the gap to a certain extent by suggesting a reasonably complete set of necessary and sufficient local optimality conditions obtained as a generalization of the classical second-order conditions in terms of the corresponding Clebsch and Riccati inequalities and the maximum principle. To this end, we use a special construction (known as the method of multiple maxima [3–5]) of the Krotov function in the Krotov sufficient conditions.

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

Pontryagin functions for multiple integral control problems

An optimal control problem is considered whose performance index is represented by anm-fold multiple integral, the state equations being given by a system of first-order partial differential equations. The concept of a field of optimal control variables with respect to an independent integral is introduced, the significance of these fields being due to the fact that a control pair is optimal whenever it is imbedded in such a field. Since there arem distinctm-fold independent integrals, it is possible to constructm distinct fields of this kind. For each of these, a Pontryagin function is defined, and it is shown that, if an optimal pair is embedded in one of these fields, it satisfies a corresponding Pontryagin maximum principle.