This paper proposes a closed form solution to L₂-sensitivity minimization of second-order state-space digital filters subject to L₂-scaling constraints. The proposed approach reduces the constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. Furthermore, restricting ourselves to the case of second-order state-space digital filters, we can express the L₂-sensitivity by a simple linear combination of exponential functions and formulate the L₂-sensitivity minimization problem by a simple polynomial equation. As a result, L₂-sensitivity is expressed in closed, form, and its minimization subject to L₂-scaling constraints is achieved without iterative calculations.