Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

Figure 3

Stationary joint probability density 𝑝(𝑎,𝛾) of the optimally controlled system (5.1) in primary external resonance case. 𝜔₀=1.0, Ω=1.1,  𝛼=1.0, 𝐸=0.3, 𝛽_𝑖=0.05, 𝜔_𝑖=30, 𝐷₁=10, 𝐷₂=5, 𝑢₀=0.1 (𝑖=1,2). (a) Theoretical result; (b) Monte Carlo simulation of the original system (5.1); (c) stationary marginal probability density 𝑝(𝑎); (d) stationary marginal probability density 𝑝(𝛾). — theoretical results; ● results from Monte Carlo simulation.

(a)

(b)

(c)

(d)