Existence and Approximation of Solutions to an Infinite Set of Linear Time-Invariant Differential Equations
Leonard L. Shaw · SIAM Journal on Applied Mathematics · 1972
Sufficient conditions are given for the existence, uniqueness and finite-dimensional approximation of the solution to a first-order infinite-dimensional vector differential equation. Given a suitable initial vector and a time-invariant system matrix, a solution exists if there is a common finite bound for the sums of the absolute values of the elements in each row of the matrix. This condition also suffices for the uniform convergence of finite-dimensional approximations. A further similar summability condition on the columns of the system matrix assures that the infinite-dimensional vector solution has a finite norm, and is unique. These results are derived by using uniform convergence properties and changes of orders of summation to justify the natural extension of finite-dimensional solutions based on the infinite series for $\exp ( At )$. An example is given of a matrix which satisfies the present conditions, but which does not have a finite norm.