T. Tadumadze^{∗} and I. Ramishvili^{∗∗}

∗I. Vekua Institute of Applied Mathematics, Tbilisi, Georgia ∗∗Georgian Technical University, Tbilisi, Georgia

FORMULAS OF VARIATION OF SOLUTION FOR QUASI-LINEAR CONTROLLED NEUTRAL DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS AND CONTINUOUS

INITIAL CONDITIONS

Formulas of variation of solution play an important role in proving necessary conditions of optimality for optimal problem [1].

In the present work, formulas of variation of solution for the neutral controlled differential equation

ẋ(t) = ∑_{j=1}^{k} A_{j}(t)ẋ(η_{j}(t)) + f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …, u(θ_{ν}(t)))

with the discontinuous (continuous) initial condition

x(t) = φ(t), t < t_{0}, x(t_{0}) = x_{0} ( x(t) = φ(t), t ≤ t_{0} )

are obtained.

Finally we note that the formulas of variation of solution for various classes of neutral differential equations are given in [2]–[6].

References

1. R. Gamkrelidze, Principles of optimal control theory. Mathematical Concepts and Methods in Science and Engineering, Vol. 7. Plenum Press, New York-London, 1978.

2. N. Gorgodze and T. Tadumadze, Differentiability of solutions of neutral type quasi-linear differential equations with respect to the initial data and righthand side. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 13(1998), No. 3, 17–20.

3. G. Kharatishvili, T. Tadumadze, and N. Gorgodze, Continuous dependence and differentiability of solution with respect to initial data and right-hand side for differential equations with deviating argument. Mem. Differential Equations Math. Phys. 19(2000), 3–105.

4. N. Gorgodze, Local Representations for the variations of solutions of some class neutral differential equations. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 15(2000), No. 3, 58–61.

5. I. Ramishvili, Formulas of variation of solution for quasi-linear controlled neutral differential equations. Mem. Differential Equations Math. Phys. 29(2003), 156–159.

6. T. Tadumadze and I. Ramishvili, Formulas of variation of solution for quasilinear controlled neutral differential equations with continuous initial condition. Georgian Math. J. 11(2004) (to appear).