We study quantum controllability under finite control resources using a multiobjective optimization perspective. By numerically constructing the feasibility boundary (Pareto front) in the resource space spanned by pulse energy and control duration, we identify the physical factors that determine this boundary. In particular, we examine the shortest control duration $T^{*}$ extracted from the boundary, below which the target fidelity cannot be achieved even with unbounded pulse energy. Using representative two- and three-level examples, we show that a finite $T^{*}$ can originate from diagonal coupling terms, small energy gaps, and relative-phase requirements in wavepacket formation, and that these limitations are not necessarily captured by conventional quantum-speed-limit arguments.

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• Quantum control