Is there a reward or penalty for validators in proof-of-stake (PoS)? In PoW consensus, if a miner could solve the puzzle, they receive some monetary reward (some coins) to incentive miners, since mining process has a considerable monetary cost for miners including providing hardware and energy/electricity cost.

However, in PoS, participation in transactions validation has not a significant monetary cost for validators (i.e. stakeholders) (I hope I am right), and if so, is there any incentive to motivate stakeholders for participating in transactions validation? 

Also, what is the penalty of a validator who ex. confirm a double-spent transaction?

Thanks
To give an example i know, in Tezos which has a variant of POS called LPOS (liquid proof of stake), at high level and without giving too much details:


the incentive to participate is offered by block (and endorsement) rewards which act de facto as a form of inflation
in order to keep bakers in check to avoid bad behavior (confirm bad transactions, double baking etc...) there's a bond/penalty system where the network locks some amount of your capital (the bond which is equal to 32x the block reward) for a period of time (2 weeks in practice). During this period if some other node can show evidence (through the protocol) that the baker was behaving bad then the bond is forfeited. 


More details about LPOS is available in this medium article

https://medium.com/tezos/liquid-proof-of-stake-aec2f7ef1da7