Controlling multiple wave properties simultaneously poses a key challenge in coherent control of wave transport. We present a theory for joint coherent control of transmission, reflection, and absorption in linear systems. We prove that the numerical range provides the mathematical structure governing achievable responses, and reveal non-Abelian effects due to noncommutativity between transmission, reflection, and absorption matrices. We provide an algorithm to achieve arbitrary target responses. Our results establish a theoretical foundation for joint coherent control of waves.

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• Light propagation, transmission & absorption
• Transport phenomena
• Wave scattering
• Complex media
• Coherent control
• S-matrix method in transport