Pointwise feedback stabilization of an unstable cascaded heat-heat system with different reaction coefficients

The present paper considers a cascade of heat-heat system with different reaction coefficients, where the actuator is presented at the arbitrary position ξ in the domain $ (0,1) $ with pointwise feedback heat flux. A sufficient and necessary condition regarding two reaction coefficients of the cascaded heat-heat model is derived to guarantee the controllability of the cascaded system. Based on the modal decomposition approach, a pointwise feedback controller is introduced to stabilize the unstable coupled system if ξ∈(0, 1) and satisfies ξ≠2k+1/2n₂ for any integers n₂∈ N⁺, k∈\0,1,2,⋯,n₂-1\. Via the Lyapunov method, sufficient conditions are derived for guaranteing exponential stability and well-posedness of the closed-loop systems. Numerical simulation demonstrates that the proposed control scheme effectively stabilizes the performance of the cascaded system.