The paper considers functional-di erential equations of pointwise type (FDEPT) which includes di erential equations with delay, that lead to solution and both at once. In the paper, we propose a technology for solving boundary value problems for nonlinear FDEPT systems which is based on the Ritz method and spline collocation approaches. To solve the problem, we discretize system trajectories on the constant step grid and formulate the generalized residual functional, including both weighted residuals of the original di erential equation and residuals of boundary conditions. Results of computational experiments for FDEPT make it possible to verify the e ectiveness of the proposed technique for investigating problems.
so on.
However, it is not applicable to functional-di erential equations of pointwise type, which, in particular, arise as the Euler-Lagrange equations for the optimal control problem with retarded argument. Studying of di erential equations of this type one can nd two diametrically opposite approach.
In the rst case, these equations are considered as a fundamentally new class according to the class of ordinary di erential equations. So, all the diversity of new properties of such equations solution are investigated.
The second way is to consider these equations as an extension of ordinary di erential equations class. This approach is based on the study of di erent characteristics, as for equations of advanced-retarded type well-known properties of ordinary di erential equations are violated. These properties include: existence and uniqueness of the solution on a given class of functions; continuous solution dependence of initial and boundary conditions; pointwise completeness of the solution; n-parametric of the solution space; \smoothness" of the solution; the \robustness" property of the equation, etc. For equations of advanced-retarded type there are nonimproving conditions under which the extension of this class of problems is possible [Beklaryan L., 2007].
In this paper, we propose an approach to the study of nonlinear functional-di erential equations, including equations with deviating argument of various kinds [Beklaryan A., 2013]. 2
Let's consider the basic initial-boundary value problem (1) (2) (3) (4) with boundary conditions and initial condition x_ (t) = f (t; x(q1(t)); :::; x(qs(t))); t 2 B;
x_ (t) = (t); t 2= B; x(t) = x; x 2 Rn; t 2 R; where B = [t0; t1], R = [t0; +1], and function qj (t), j = 1; s are homeomorphisms of straight line that maintain the orientation. In general, when the deviation of an argument is arbitrary, we have a problem with nonlocal initial boundary conditions.
All speci c FDEPT features emerge because the Lipschitz constant loses duties that it performs for the ODE.
Let L be Lipschitz constant for right side of FDEPT [Beklaryan L., 2007] f ( ), hi = maxt2R jqi(t)j, i = 1; s. For > 0 de ne an inequality
s L X exp jhij < :
i=1 Obviously, hi = 0, i = 1; s conditions is true for ODE. For this equation there exists such > 0, that for all 2 ( ; +1) inequality (4) is satis ed.
If the equation is not an ordinary di erential equation, then inequality (4) either does not have a single solution with respect to > 0, or it is found 2 > 1 > 0 that inequality (4) is satis ed for any 2 ( 1; 2).
Theorem [Beklaryan L., 2007]. If inequality (4) holds for some > 0, then for the initial boundary-value problem (1){(3) there exists a solution whose asymptotic behavior is upper-majorized by an exponential function. Such solution is unique.
Consequence. Under the hypotheses of the theorem, for any arbitrarily small " > 0 a solution exists in the class of functions asymptotically majorized by an exponential function " 1+"jtj, and it is unique in the class of functions asymptotically majorized by an exponential function " 2 "jtj.
By the above theorem, if we con ne ourselves to solutions with the asymptotic behavior indicated in the theorem, then for them all the properties of the ODE solutions are satis ed: the existence and uniqueness of the initial-boundary value problem; the continuous dependence of the solution on the initial-boundary conditions; nparametrization of the solution space; pointwise completeness of solutions; assignment of motion along solutions as homeomorphisms of phase space.
In the paper, it is proposed to reduce the initial-boundary problem presented to the optimization problem for the purpose of further investigation of it by applying the developed computing technology.
Dynamics of the system on the main time interval is described by equations
0 = Fi(x(g(t)); x_ (g(t))); t 2 [t0; t1]; i = 1; dim n; where Fi : Rdim n Rdim n ! R1. On the extended interval of the independent variable variation t 2 [tN ; tK ], tN t0, tK t1, outside the basic interval, the values of the phase variables derivatives are de ned x_iL = hiL(t), t 2 [tN ; t0] and x_ iR = hiR(t), t 2 [t1; tK ], i = 1; dim n. Phase variables on the extended time interval must satisfy constraints xli xi(g(t)) xgi, i = 1; dim s. Functions of homeomorphisms gi(s), i = 1; dim s are de ned on the interval s 2 [0; 1] and must satisfy the monotonicity conditions dgi > 0; i = 1; dim s: dsi The boundary conditions are given by the functionals Kj (x(g( j )); x_ (g( j ))) = 0, j = 1; dim l, j 2 [t0; t1]. Thus, the dimensions of a problem are de ned by integer variables dim n, dim s, dim r, dim l. All the elements of the mathematical formulation of the problem must satisfy the corresponding natural conditions of smoothness and consistency.
To solve the presented problem it is formulated the convolution of discrepancies functionals I(x(t)) = Pid=im1 n viN RttN0 [x_ i(g(t)) hiL(t)]2dt+ Pid=im1 n viN Rtt1K [x_ i(g(t)) hiR(t)]2dt+
Pjd=im1 l vjK Kj2(x(g( j )); x_ (g( j ))) + K0(x(g( 0)); x_ (g( 0))); where viN , i = 1; dim n and vjK , i = 1; dim l are weighting coe cients for discrepancies and boundary conditions. 4
The proposed technology for study boundary value problems is based on the Ritz method and spline-collocation approaches. To solve the problems of the class under consideration, the trajectories of the system are discretized on a grid with a constant step and a error functional is formulated. It includes both the weighted discrepancies of the original di erential equation and the discrepancies of the boundary conditions. To evaluate the derivatives of the system trajectories, the \spline di erentiation" technique is used, based on two spline approximation methods: using cubic natural splines and a special type of splines whose second derivatives at the edges are also optimized.
To solve the set nite-dimensional optimization problems, in the general non-convex case, a set of local
optimization algorithms is implemented
The corresponding software OPTCON-F is implemented in C-language under the control of operating systems OS Windows, OS Linux and Mac OS using compiler GCC and is intended for numerical solution of boundary value problems, parametric identi cation problems and optimal control problems for dynamical systems described by the functional-di erential equations of pointwise type.
Algorithmic lling of the complex includes a set of local (unimodal) and global (nonlocal) optimization algorithms. The techniques can be divided also into basic algorithms (optimization from any initial approximation) and algorithms for re nement of the taken result. The duration of the algorithm is limited to either the speci ed number of iterations, or the achievement of one of the stopping criterions (the minimum gradient norm for the current point is less than a given threshold value or the algorithm (for given values of algorithmic parameters) can not nd a better approximation. Among the local algorithms included in OPTCON-F are: Parthan method; the classical Powell-Brent method; the gradient method of trust region; Barzilai-Borwein method; Newton's method with a di erence estimate of the Hessian matrix; generalized quasi-Newton method; direct-dual method (5) (6) (7) of gradient search; di erential Euler optimization method of the second order; di erential optimization method of Adams of the fourth order and others. As algorithms for recti cation results obtained at the rst stage are adaptive modi cation of Hooke-Jeeves method, stochastic search methods in random subspaces of the indicated dimension (2, 3, 4, or 5), local version of the curvilinear search method. Nonlocal algorithms in the basis of the OPTCON-F are \parabol method" (a combination of coordinate-wise search with a multistart and a nonlocal one-dimensional search using the parabola algorithm), nonlocal method of curvilinear search, the Luus-Yaakola method, the \forest method" (a multivariant adaptive method of random multi-start) and others. 5
A fragment of the computational experiments for boundary value problems for FDEPT systems using the developed computing technology is presented in this paragraph. In addition to analyzing the stability of the proposed numerical solutions for initial-boundary conditions, their main properties have also been studied, in particular, the qualitative properties of the majorant of such solutions have been investigated. 5.1
The initial-boundary test problem 01 is formulated as follows x_ 1(t) = x2(t2); x_ 2(t) = 4x1(t + 1) 2x2(t x_ 1(t) = 0; x_ 2(t) = 1; t 2 [ 1; 0) [ (1; 2]; x1(0) = 1; x2(0) = 2:
This system consists of equations with argument deviations of complicated structure. The graphs with solution of this test problem obtained by OPTCON-F are shown in Fig. 1. (8) (9) 5.2
We consider the boundary-value problem in the following formulation x_ 1(t) = 12 x2(t) 12 x2(t + 1); x_ 2(t) = 12 (t x_ 1(t) = 4; x_ 2(t) = 1; t 2 [ 1; 0); x_ 1(t) = et; x_ 2(t) = et; t 2 (1; 2]; x1(0) = 1; x2(0) = 2: The dynamical process in test problem 03 is described by the following system of di erential equations with deviating argument x_ 1(t) = sin(t); x_ 2(t) = x1(t + 1) (t + 1)x2(t x_ 1(t) = sin(t); x_ 2(t) = sin(t); t 2 [ 1; 0); x_ 1(t) = cos(t); x_ 2(t) = cos(t); t 2 (1; 2]; x1(0) = 2; x2(0) = 1:
The results of the numerical solution of the test problem 02 are shown in Fig. 3. The minimum value of the convolutional functional corresponding to the system (10) with a grid of 241 nodes is 0:00074. The results of computational experiments for FDEPT with deviating argument in the right-hand parts of equations, make it possible to verify the operability of the proposed technology for investigating problems of this type. In the future, it is planned to develop the presented ideas for solving optimal control problems for systems with di erential coupling of the FDEPT type. This work is partially supported by Russian Foundation for Basic Research, project No. 17-07-00627.