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We solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in a contact space. Pontryagin’s maximum principle is used to find an optimal trajectory (thermodynamic process) on this manifold that maximizes the work of the gas. In the case of ideal gases, it is shown that the corresponding Hamiltonian system is completely integrable and its quadrature-based solution is given. : contact geometry, thermodynamics, optimal control, Hamiltonian systems, integrability

The theory of finite dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depend on a finite number of parameters among all solutions of evolutionary differential equations.

This article proposes an approach to constructing exact solutions of a third-order nonlinear evolutionary differential equation based on the theory of finite-dimensional dynamics. This theory, in turn, is based on the theory of shuffling symmetries of ordinary differential equations and it is a natural extension of the theory of dynamical systems to evolutionary partial differential equations. theory of finite-dimensional dynamics makes it possible to find families of solutions of evolution equations, depending on a finite number of parameters, among all solutions of such equations.

We consider the problem of two-phase filtering (oil and water) in a horizontal layer of an oil deposit. We propose an asymptotic method for calculating both the filtering process and related optimal control problems, namely the problems of choosing optimal control actions to achieve maximum oil production at a given level of water resource consumption or a minimum water flow rate that provides the level of oil production required by the plan.

The nonisothermal two-phase porous media flow of oil and hot water in a horizontal oil reservoir is considered. An asymptotic method for calculating the flow and solving related optimal control problems is proposed. Namely, the problem of choosing optimal control actions to maximize the oil production for a given level of financial costs and to minimize the costs for a given level of oil production is considered.

In this article we study the initial boundary value problem for two-phase heat and mass transport in porous media described by the Buckley–Leverett model. We outline a method to construct asymptotic solutions of the initial boundary problem and show how to overcome singularities in solutions and shock waves.

A numerical method is proposed to construct attractors of evolution equations describing the process of filtration of a two-phase liquid with allowance for capillary forces (the Rapoport-Leas model). The stability conditions for the proposed method are the conditions under which the third-order dynamics is an attractor for the initial equation. The solution of the initial equation is an iterative process, which can be paralleled to accelerate calculations.

For the third order evolution differential equations, the algebra of differential invariants with respect to the pseudogroup of point transformations is constructed. Such equations arise in the study of filtration processes (the Rapoport – Leas equation), the theory of nonlinear waves (the Korteweg – de Vries equation).

Finite-dimensional lower-order dynamics are constructed for evolutionary differential equations generalizing the well-known Burgers – Huxley equation. These dynamics allow us to construct exact solutions to such equations. Some of them are presented in this paper.

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential equations by B. Kruglikov, O. Lychagina and V. Lychagin. This theory is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of PDEs. The proposed method is used to construct exact particular solutions of linear differential equations (Helmholtz equations and equations of variable type).

A constructive method is proposed for finding finite-dimensional submanifolds in the space of smooth functions that are invariant with respect to flows defined by evolutionary partial differential equations. Conditions for the stability of these submanifolds are obtained. Such submanifolds are constructed for generalized Rapoport–Leas equations that arise in the theory of porous media flows.

For the Buckley–Leverett equation describing the flow of two immiscible fluids in porous media, an exact parametric representation of the solution is constructed with the help of the Backlund transformation. As a result, the advance of the displacement front can be controlled to a high degree of accuracy. The method is illustrated using an example of a typical oil well with actual parameters.

In this paper we consider the Buckley-Leverett filtration model for a two-phase system (for instance, water oil) in a plane domain of porous media with injection and producing wells. Buckley-Leverett’s equations supplemented by the law of conservation of thermal energy Asymptotic solutions for two-phase flows in porous media are studied. Dimensionless numbers for characterization of these flows are introduced. These numbers we use to construct asymptotic presentations of solutions. We give explicit formulae for the initial terms of this asymptotic which we use for optimal control of oil-field development.

According to the results of B. Kruglikov, O. Lychagina, and V. Lychagin, evolutionary partial differential equations determine flows (finite-dimensional dynamics) on solutions’ spaces of ordinary differential equations. In the present paper, we construct such dynamics for the classical Burgers–Huxley equation and then we use them to construct new exact solutions.

The main goal of these lectures is to give a brief introduction to application of contact geometry to Monge–Ampere equations.

A differential-geometric description of the models of two-phase filtration in a plane anisotropic porous media is proposed. The concept of the anisotropy metric is introduced. A generalization of the Buckley-Leverett model for anisotropic media is constructed.

A differential-geometric description of the models of multiphase 3-D filtration in anisotropic porous media is proposed. The concept of the anisotropy metric is introduced. A generalization of the Buckley-Leverett model for anisotropic media is constructed.

One-dimensional Buckley - Leverett’s model of frontal displacement of petroleum by water is considered. We apply Backlund transformation to the Buckley - Leverett equation. This allows us to find Cauchy problem’s solutions for this equation in quadratures. These results can be used in control of development of oil fields.

One-dimensional models of frontal displacement of petroleum by active reagents are considered. These models are described by nonlinear systems of two first order partial differential equations — so called Jacobi systems (see Lychagin (1992); Kushner, Lychagin, Rubtsov (2007)). This article can be considered as a continuation of our work on the application of differential geometric methods in control theory the development of oil fields (see Akhmetzianov, Kushner, Lychagin (2013,-) and Kushner, Lychagin, Rubtsov (2007); Lychagin (1992, 1993)).

Time-optimal management of the development of oil fields discussed. We use one-dimensional Buckley-Leverett filtering model.

Optimal strategy of development of oil fields is constructed for one-dimensional the Buckley-Leverett model. The obtained results are intended for use in field development conventional or heavy oil alternating parallel rows of vertical production and injection wells and alternating trunks or vertical cracks hydraulic fracturing in the bottom-hole zones of horizontal production and injection wells.

Asymptotic solutions for two-phase flows in porous media are studied. Dimensionless numbers for characterization of these flows are introduced. These numbers we use to construct asymptotic presentations of solutions. We give explicit formulae for the initial terms of this asymptotic.

The present paper is the first in the projected series of publications illustrating the application of the geometric theory of singular solutions of the nonlinear partial differential equations to the description of special modes in the distributed-parameter control systems. Consideration was given to the problem of biphase unidimensional filtering of fluids (oil and water) in the porous media of the natural oil pools. The paper is an extended version of the report presented at the International Conference PACO'2012 prepared for publication on recommendation of the Program Committee [1].

Source at https://doi.org/10.1016/j.procs.2017.08.038. Accepted manuscript version, licensed CC BY-NC-ND 4.0. We construct new finite dimensional submanifolds in the solution space of nonlinear differential filtration equations and describe the corresponding evolutionary dynamics. This method is implemented in a computer program of symbolic computations Maple.

A method of constructing finite dimensional invariant submanifolds (= dynamics) for second order evolutionary differential equations is described. This method allows one to construct attractors of such equations and investigate stability of their solutions.

The classification problem of general control Hamiltonian systems with respect to feedback transformations is considered. The Poisson structure on the algebra of differential invariants is constructed. For systems with one control parameter we find a solution of equivalence problem.

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampere equations and Jacobi equations.

We solve a problem of local contact equivalence of hyperbolic and elliptic Monge-Ampere equations to linear equations with constant coefficients. We find normal forms for such equations: the telegraph equation and the Helmholtz equation.

A distinguishing feature of suchequations in the class of all second-order equations isthat it is closed with respect to contact transformations.This fact was known even to Sophus Lie, who studiedMonge–Ampere equations by methods of contactgeometry, which he created. In the 1870s and the1880s, Lie posed the problem of classifying theMonge–Ampere equations with respect to the(pseudo)group of contact transformations and, in par-ticular, reducing Monge–Ampere equations to quasilin-ear form (corresponding to D = 0 in Eq. (1)) and findingthe simplest coordinate representation of such equa-tions [8, 9].He also considered classification problems forMonge–Ampere equations admitting intermediate inte-grals. Lie himself found conditions for reducingMonge–Ampere equations to quasilinear form and tolinear equations with constant coefficients, but proofsof these theorems have never been published. Note,however, that verifying the presence of intermediateintegrals for general Monge–Ampere equations, muchmore constructing them, is a difficult problem.In 1983, V.V. Lychagin and V.N. Rubtsov consideredthe problem of classifying the Monge–Ampere equa-tions defined by (1) where the coefficients A, B, C, D,and E do not explicitly depend on the function v. They

We solve a problem of contact linearization for non-degenerate regular Monge-Ampere equations. In order to solve the problem we construct tensor invariants of equations with respect to contact transformations and generalize the classical Laplace invariants.

The class of Monge–Ampere equations is closedunder contact transformations, which is the propertythat singles out this class from the entire set of second-order equations. This fact was known to Sophus Lie,who studied Monge–Ampere equations by applyingcontact geometry methods [9, 10]. Lie also consideredclassification problems for Monge–Ampere equationspossessing intermediate integrals.In 1979, V.V. Lychagin proposed a geometricdescription for a wide class of second-order differentialequations on smooth manifolds. For a two-dimensionalmanifold, this class coincides with the class of Monge–Ampere equations (1). Lychagin’s basic idea [14] isthat Monge–Ampere equations and their multidimen-sional analogues are represented by differential formson the 1-jet space on a smooth manifold. An advantageof this approach over the classical one is an order reduc-tion of the jet space: the 2-jet space