Solving Boundary Value Problem for a Nonlinear Stationary Controllable System with Synthesizing Control

An algorithm for constructing a control function that transfers a wide class of stationary nonlinear systems of ordinary differential equations from an initial state to a final state under certain control restrictions is proposed. The algorithm is designed to be convenient for numerical implementation. A constructive criterion of the desired transfer possibility is presented. The problem of an interorbital flight is considered as a test example and it is simulated numerically with the presented method.


Introduction
One of the problems of mathematical control theory is developing of exact or approximate methods to construct control functions and corresponding trajectories, which connect given points in the phase space. A large amount of publications is devoted to researches in this field, for instance [1][2][3][4][5][6][7][8][9][10][11][12]. Today boundary value problems (BVPs) are quite well studied for linear and non-linear controllable systems of the special form. However the theory of BVPs for general non-linear controllable systems has not yet been sufficiently developed. The main goal of the authors was to construct an algorithm of solving BVPs for a larger class of non-linear controllable systems of ordinary differential equations in the class of synthesizing controls, which would be numerically stable and easy to implement with computer, and to find a constructive sufficient condition of the solution existence for such problems. This goal was reached by reducing the original problem to a linear non-stationary system of a special form and solving the initial value problem for an auxiliary system of ordinary differential equations. The efficiency of the presented algorithm is demonstrated with numerical simulation of a certain practical problem.
The object of the study is a controllable system of ordinary differential equations (ODEs) We say that such pair ( ), ( ) is a solution of the problem (1.1), (1.6).
Theorem. Let the conditions (1.2), (1.3) and (1.4) to be satisfied for the right-hand side of (1.1). Then ∃ > 0 such that ∀ ̅ ∈ : ‖ ̅ ‖ < there exists a solution of the problem (1.1), (1.6), which can be found after solving, first, a problem of stabilizing a linear non-stationary system with exponential coefficients and, second, an initial value problem for an auxiliary ODE system.
The main idea of the proof is to use successive changes of independent and dependent variables to reduce the process of solving the original system to the problem of stabilizing a non-linear auxiliary system of ODEs of the special form under constant perturbations. To solve the latter we find a synthesizing control, which provides exponential decrease of the linear auxiliary system fundamental matrix. At the final stage we return to the original variables.
We make a change of the independent variable in the system (2.2): where > 0 is a certain constant value to be determined. Then in terms of (2.2) and (2.4) take form: Using the property (1.2) and Taylor series expansion of the right-hand side of (1.1) about ( ̅ , 0), we can rewrite the system (2.6) as: (2.9) Let's bound the range of ( ) with ‖ ( )‖ < 1 , ∈ [0, ∞). (2.10) We will now shift the functions ( ), = 1, … , , several times. Our aim is to get an equivalent system where all the terms in the right-hand side, which do not contain powers of or in explicit form, would be of the order ( −4 ‖ ̅ ‖) as → ∞ and ‖ ̅ ‖ → 0 in the area (1.5), (2.10).
At the second stage, we make a change of variables (2) (0) = 0, = 1, … , . (2.14) With respect to these new variables the original system (2.12) and the initial conditions (2.13) are written as: Comparing to the previous variables shift, the right-hand side terms of (2.15), which do not contain the powers of the components of or in explicit form, are now of the order ( −3 ‖ ̅ ‖) as → ∞ and ‖ ̅ ‖ → 0 in the area (1.5), (2.10).
By induction, at the -th stage, using (2.11)-(2.16) we have the necessary shift of the form: (2.17) We apply (2.17) 4 − 1 times and collect the terms, which are linear in respect to the components of (4 −1) and include the coefficients − , = 1, … , , and also the terms, which are linear in respect to the components of and include the coefficients − , = 1, … ,2 . Now we have the system, which according to (2.12)-(2.17) can be written in vector form as: (2.18) The functions 1 consist all the terms which are linear in respect to the components of (4 −1) with coefficients − , ≥ + 1, and also the terms of the last sum of the right-hand side, for which | | = 0 and | | = 1. The functions 2 consist all the terms which are linear in respect to the components of with coefficients − , ≥ 2 + 1, and also the terms of the last sum of the right-hand side, for which | | = 1 and | | = 0. In 3 all the terms, which are non-linear in respect of the components of (4 −1) or , are contained. Finally, the functions 4 include all the terms, which do not have powers of (4 −1) and components. The functions and have form (2.19) (2.20)
(4.6) From (4.5) and (4.6) it follows that the condition (4.3) is satisfied in the area ‖ ̅ ‖ < 2 . Moreover, in that area the structure of 3 provides the estimation With use of (4.3), change the variable (4 −1) according to the expression (4 −1) = 2 ( ) . According to [13] for the first term of the right-hand side of (4.9) we have   Denote the fundamental matrix of the system (4.13) closed with the control (4.14) via Ψ( ), (Ψ(0) = , an identity matrix). It is obvious that the elements of Ψ( ) are exponential functions of negative argument or their derivatives.
Consider the system (3.15) with the control (4.16)

Practical example
To demonstrate the effectiveness of the proposed method we consider a problem of transferring a material point moving in a central gravitational field to a desired circular orbit with jet power.
According to [12] the system in deviations from the prescribed circular motion and the condition (1.6) take form Here 1 = − 0 , 2 =, 3 = − 0 ; 0 = ( 0 ) 1 2 , = ⁄ ; 0 is the circular orbit radius; ̇ is the radial velocity; is the generalized momentum; and are the relative velocity vector projections onto the radial and tangential directions respectively (they are constant); is the mass and ̇ is its change rate; = 0 where 0 is the universal gravitational constant; is the mass of the Earth; vector = ( 1 , 1 , 3 ) ; and the control ∈ 1 .

(6.9)
It is obvious that (6.9) provides an exponential decrease of (6.6) solutions. At the final stage we solve the initial value problem for the system obtained from (6.3) after changing the phase coordinates according to (6.5) with the control (6.9). Then we return to the original variables. The initial values for the Cauchy problem are During the numerical simulation we have solved the auxiliary system of ODEs constructed from (6.3) and (6.9) after changing the phase coordinates according to (4.5) with the initial values 1 (0), 2 (0), 3

Conclusions
The analysis of the Theorem proof shows that the most difficult and time-consuming part of the algorithm implementation can be proceeded with analytical methods of computer algebra packages. The results of the numerical simulation of interorbital flight convince that the method can be used for construction and simulation of various technical objects control systems.