On Some Linear Two-Dimensional Volterra Integral Equations of the First Kind

The problem of identifying Volterra kernels is an important stage in the construction of integral models of nonlinear dynamical systems based on the tool of Volterra series. The paper considers a new class of two-dimensional integral equations that arise when recovering nonsymmetric kernels in a Volterra polynomial of the second degree, where \( x(t) \) is the input vector function of time. The strategy for choosing test signals used to solve this problem is based on applying piecewise linear functions (with a rising edge). An explicit inversion formula is constructed for the selected type of Volterra equations of the first kind with variable integration limits. The questions of existence and uniqueness of solutions of the corresponding equations in the class \( C_{[0,T]} \) are studied.