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Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I implement is not the optimal one. Is there literature on measuring the deviation of suboptimal trajectories from optimal ones, studying properties of the system that is subject to such suboptimal trajectories?

I understand that while optimal trajectory is a well-defined object there's an infinite number of suboptimal trajectories. My question is if there is a systematic way to study those, put bounds on their suboptimality, deriving some index of optimality etc.

Thanks!

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Model predictive control (MPC, aka receding horizon control) is one type of sub-optimal control method that is extremely well studied and popular. The "sub-optimality" of this type of control methods is also studied. E.g., see

1). Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems, Automatica, 48 (2011)

2). Grüne, Lars, and Jürgen Pannek. "Practical NMPC suboptimality estimates along trajectories." Systems & Control Letters 58.3 (2009): 161-168.

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