A　relationship　between　stability　and　self-optimization　is　found　for　weakly　dissipative　heat　devices.　The　effect　of　limited　control　on　operation　variables　around　an　steady　state　is　such　that,　after　instabilities,　the　paths　toward　relaxation　are　given　by　trajectories　stemming　from　restitution　forces　which　improve　the　system　thermodynamic　performance　(power　output,　efficiency,　and　entropy　generation).　Statistics　over　random　trajectories　for　many　cycles　shows　this　behavior　as　well.　Two　types　of　dynamics　are　analyzed,　one　where　an　stability　basin　appears　and　another　one　where　the　system　is　globally　stable.　Under　both　dynamics　there　is　an　induced　trend　in　the　control　variables　space　due　to　stability.　In　the　energetic　space　this　behavior　translates　into　a　preference　for　better　thermodynamic　states,　and　thus　stability　could　favor　self-optimization　under　limited　control.　This　is　analyzed　from　the　multiobjective　optimization　perspective.　As　a　result,　the　statistical　behavior　of　the　system　is　strongly　influenced　by　the　Pareto　front　(the　set　of　points　with　the　best　compromise　between　several　objective　functions)　and　the　stability　basin.　Additionally,　endoreversible　and　irreversible　behaviors　appear　as　very　relevant　limits:　The　first　one　is　an　upper　bound　in　energetic　performance,　connected　with　the　Pareto　front,　and　the　second　one　represents　an　attractor　for　the　stochastic　trajectories.