Periodic media offer rich wave-controlling capabilities for diverse applications including seismic mitigation, noise control, and energy harvesting. This work presents an inverse-design framework to tailor energy velocity and band gaps through complex band structure analysis, providing a more efficient approach for engineering wave-based devices. Traditional band structure calculations solve for frequency given wavenumber, yielding a linear Hermitian eigenvalue problem with real eigenvalues. While this real band structure reveals fundamental wave characteristics, it is inconvenient for design purposes where decisions typically target specific frequencies. For example, identifying band gaps requires sweeping the entire wavenumber space and tracking neighboring band limits, making the process computationally demanding. Complex band structure analysis reverses this approach by solving for complex-valued wavenumber given frequency, forming a quadratic eigenvalue problem. This formulation sharply identifies band gaps as frequency ranges where wavenumbers become complex-valued, requiring only one eigenvalue analysis per frequency of interest. Thus, complex band analysis is ideally suited for inverse design problems. We introduce two complementary design approaches based on complex band structure: a discriminant-based method that opens band gaps precisely at user-specified frequency ranges, and a group-velocity-based method for controlling energy velocities at target frequencies. Both approaches utilize local information from eigenpairs at specific frequencies rather than global band structure data. The inverse design problem is formulated as a constrained minimization with the Floquet-Bloch eigenvalue problem serving as the constraint. Gradient-based optimization iteratively determines optimal material properties or geometrical parameters. The framework is demonstrated for both acoustic and elastic wave equations, with time-domain verification using finite unit cell assemblies. This methodology provides a systematic approach for designing periodic metamaterials with prescribed wave-controlling capabilities across various engineering applications.