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We address the problem of optimizing the performance of a dynamic systemwhile satisfying hard safety constraints at all times. Implementing an optimalcontrol solution is limited by the computational cost required to derive it inreal time, especially when constraints become active, as well as the need torely on simple linear dynamics, simple objective functions, and ignoring noise.The recently proposed Control Barrier Function (CBF) method may be used forsafety-critical control at the expense of sub-optimal performance. In thispaper, we develop a real-time control framework that combines optimaltrajectories generated through optimal control with the computationallyefficient CBF method providing safety guarantees. We use Hamiltonian analysisto obtain a tractable optimal solution for a linear or linearized system, thenemploy High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) toaccount for constraints with arbitrary relative degrees and to track theoptimal state, respectively. We further show how to deal with noise inarbitrary relative degree systems. The proposed framework is then applied tothe optimal traffic merging problem for Connected and Automated Vehicles (CAVs)where the objective is to jointly minimize the travel time and energyconsumption of each CAV subject to speed, acceleration, and speed-dependentsafety constraints. In addition, when considering more complex objectivefunctions, nonlinear dynamics and passenger comfort requirements for whichanalytical optimal control solutions are unavailable, we adapt the HOCBF methodto such problems. Simulation examples are included to compare the performanceof the proposed framework to optimal solutions (when available) and to abaseline provided by human-driven vehicles with results showing significantimprovements in all metrics.

Bridging the Gap between Optimal Trajectory Planning and Safety-Critical Control with Applications to Autonomous Vehicles