The monograph consists of three parts. Chapter 2–5 of Part 1 give the detailed overview of the theory behind the continuous-time Volterra models of evolving dynamical systems, beginning in Chapter 2 with Volterra integral equation (VIE) of the first kind with piecewise continuous kernels. The algorithm for construction of the continuous solutions in the form of logarithmic-power asymptotics is presented, the coefficients of which \(x_{i}(\ln(t))\) are constructed as polynomials on powers of \(\ln (t)\) and may be dependend on certain number of constants. The existence theorems separately for regular and singular cases are proved. In Chapter 3, the matrix VIEs of the first kind with jump discontinuous kernels are studied. Again, the algorithm is presented for the construction of the logarithmic-power asymptotics of the unknown continuous solutions. Regular and singular cases are studied according to the characteristic equation roots.
The existence and uniqueness theorem is proved together with sufficient conditions for the solution existence. The results of the same type are obtained in Chapter 4 for Volterra operator equations (VOE) of the first kind with piecewise continuous kernels, however, the solution of the VOE in the nonregular case belongs to the class of distributions in the sense of Sobolev-Schwartz theory and its construction is realized in Chapter 5.
In Part 2, the author studies nonlinear continuous-time integral models beginning in Chapter 6 with the nonlinear Hammerstein integral equation with a single parameter and extends the suggested technique on the abstract operator equations with vector parameter. In Chapter 7, the nonlinear Volterra integral equations with non-invertible operator in the main part are considered with the aim to obtain sufficient conditions for the existence of solutions in a neighborhood of the algebraic branching point. The asymptotics values of solution branches are constructed and a successive approximation method is indicated to be uniformly convergent in their neighborhoods. When the VIE hasn’t continuous solutions the method for construction of generalized solutions in the Sobolev-Schwartz distributions space. In Chapter 8, an algorithm is suggested for the construction of parametric families of small branching solutions of nonlinear differential equations of \(n\)th order in the neighborhood of the branching point and is demonstrated on the example of nonlinear DEs arising in magnetic insulation problem.
In Chapters 9, 10, 11 the author studies classic and generalized solutions to nonlinear Volterra equations arising in the theory of nonlinear continuous-time casual dynamical systems control based on the Volterra integral-functional series. For the proof of the continuous solution existence theorem in Chapter 9, a convex majorants method is developed. Generalized solutions are studied in Chapter 10. Chapter 11 is devoted to impulse control on nonlinear dynamical systems based on the Volterra series.
Part 3 of the monograph is devoted to applications of integral models. In Chapter 12, Volterra series based models are applied to nonlinear heat-exchange dynamics modeling. Chapter 13 “Suppression of Moire patterns for video archive restoration” presents the practical examples of the Fourier transform usage in digital signal processing application and particularly in motion pictures restoration. In Chapter 14, the integral models applications in electric power engineering are presented, particularly in Section 14.1 – the Hilbert-Huang integral transform usage is to electric power system parameters forecasting and in Section 14.1 – nonstationary autoregressive model for on-line deflection of inter-area oscillations in power systems.