Existence of a unique solution of an infinite three-coupled system model of ordinary differential equations in Hilbert space

Numerous practical problems are often formulated as control problems modeled by partial differential equations (PDE). The spectral decomposition method often transforms such problems into an equivalent control problem modeled by an infinite system of ordinary differential equations (ODE). We examine a solution to an infinite system of three-coupled ODE in the Hilbert space, ℓ2. The existence and uniqueness of the solution to the given infinite system in the Hilbert space ℓ2 are proved as our main research findings. Our findings imply that control and differential games modeled by this infinite system of three-coupled ODE can now be investigated.

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