During chronic viral infection, sustained antigen stimulation leads to exhaustion of virus-specific ${\rm CD}8^+$ ${\rm T}$ cells, characterized by elevated expression of inhibitory receptors and progressive functional impairment, including loss of cytokine production, reduced cytotoxicity, and diminished proliferative capacity. In this paper, to investigate how ${\rm T}$ cell exhaustion influences viral persistence, we developed a within-host mathematical model integrating viral infection dynamics with adaptive immune responses. The model demonstrates three non-trivial equilibria: infection-free equilibrium ($S_1$), uncontrolled-infection state ($S_2$), and immune-controlled equilibrium ($S_3$). Through dynamical systems analysis, we established the local stability of all states ($S_1$-$S_3$) and prove global stability for both $S_1$ (complete viral clearance) and $S_2$ (chronic infection). Notably, the system exhibits Hopf bifurcations at $S_2$ and $S_3$, with distinct critical thresholds governing oscillatory dynamics. Numerical simulations reveal that successful immune-mediated control of viral load and infected cell levels requires maintenance of low ${\rm CD}8^+$ ${\rm T}$ cell exhaustion rates.