A new approach to the method of nonlinear variation of parameters

Capture in Nonlinear Differential Games of Pursuit

“…The topics studied here are the non-linear differential equation analogues of these fundamental results of Hajek [7] and Chukwu [8], the first and second reciprocity theorems in [7] and G-controllability in [8]. The thrust of our result is that the non-linear pursuit problem (nonlinear control systems under square integrable but unpredictable perturbations) is reduced to one in control theory.…”

“…Furthermore, , = + modulo ker Φ 1 , , , for all ∈ , ∈ 0 , 1 ,can be used to determine a suitable control strategy from an admissible control ∈ 2 ( 0 , 1 , ) for (74) and vice versa. This is the nonlinear analogue to Hajek's [7] first duality theorem. We shall omit the proof of Proposition 3.1since it is analogous to the more general situation in Proposition 3.2.…”

On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities

“…Due to the form of the required system (1) that modulates a perturbation of phenomena registered by a time-varying dynamical linear system (2), there is an explicit form of solutions obtained in the same way as those for the discrete and continuous cases given by Alekseev's formula in 1961 (see [1], or some of its generalizations [26]). Thus, we are going to utilize the recording time scale version of the variation of parameters formula, shown in [8, section 3], to analyze stability estimations.…”

“…, and x(t) := x(t, t 0 , x 0 ) be a solution of system (1) that satisfies the variation of constant formula (26).…”

A nonlinear variation of constants method for integro differential and integral equations

“…Oun next nesult extends a differential equations resul"t of Lord and l,litchetl [15] whene a{trtoru{t)) is a solution of (4"1) u(t) is a solution of (?,5), (Recal- If ne let z,(trt*ur), * ? *t, arrd q(*ntrru/ r to r.t S t, be the maximal solutions of (4"9) and (4.10) nespeetively then the theony of diffenential lnequalities together' *rith (4"11-) …”